I can't edit the opening post anymore, so here's a revised list, which I've tidied up a little and added some more information.
I've also retouched the definitons a little, so their intent becomes clearer.
What species are currently mostly or entirely inaccessible to world birders?
There are four categories:
A) Birds whose populations / exact occurrences are unknown or not understood.
B) Birds that occur in inaccessible locations that would require concerted, self-organized expeditions because they cannot be reached otherwise.
C) Birds that occur only in inaccessible areas that require a complicated permitting process, such as protected areas that can only be visited by certain researchers.
D) Birds that are currently inaccessible because they occur only in politically unstable regions, such as regions of war or terror.
Here is a possible, preliminary list. Feel free to add more!
A)
1215 Dulit Partridge (Rhizothera dulitensis); No confirmed reports since 1937, known from Borneo
1298 Edwards's Pheasant (Lophura edwardsi); No confirmed sightings of wild birds since 1996, known from Vietnam
1634 New Caledonian Nightjar (Eurostopodus exul); Known only from a single specimen from New Caledonia
1880 Nechisar Nightjar (Caprimulgus solala); Known only from a single specimen found in 1990 in Ethiopia, of which only one wing was preserved
1902 Prigogine's Nightjar (Caprimulgus prigoginei); Known only from a single specimen from DRCongo
1992 New Caledonian Owlet-Nightjar (Aegotheles savesi); Known only from two specimens, a probable sighting in 1998 and a handful of unconfirmed reports from New Caledonia
2601 Letitia's Thorntail (Discosura letitiae); Known only from two or three questionable specimens from the 19th century, presumably from Bolivia
4324 Purple-winged Ground Dove (Paraclaravis geoffroyi); Very poorly understood and extremely rare nowadays, occurs in Brazil, Argentina and Paraguay
4955 Colombian Crake (Neocrex colombiana); Very poorly understood though possibly not rare locally, occurs in Colombia, Panama and Ecuador, possibly Costa Rica
4959 Zapata Rail (Cyanolimnas cerverai); Very poorly understood and probably rare, occurs on Cuba
5063 Brown-banded Rail (Lewinia mirifica); Only known as passage migrant in Luzon, Phillippines and likely declining
5089 New Caledonian Rail (Cabalus lafresnayanus)
5166 Makira Woodhen (Gallinula silvestris)
5540 Worcester's Buttonquail (Turnix worcesteri); Poorly understood, rarely encountered (as migrant?) in Luzon, Phillippines
5847 Eskimo Curlew (Numenius borealis); Almost certainly extinct
5850 Slender-billed Curlew (Numenius tenuirostris)
6058 Jerdon's Courser (Rhinoptilus bitorquatus); Until recently, considered extinct, rediscovered in India 1986, subsequently lost again.
7549 Chestnut-shouldered Goshawk (Erythrotriorchis buergersi); Very poorly known, records from its range in New Guinea range very scarce
8034 Itombwe Owl (Tyto prigoginei); Very poorly understood, likely declining in its range in DRCongo (plus Ruanda and Burundi?)
8460 Maned Owl (Jubula lettii); Very poorly known, records from its range in central Africa very scarce
8512 Shelley's Eagle-Owl (Ketupa shelleyi); Very poorly known, records from its range in central Africa very scarce
9457 Sangihe Dwarf Kingfisher (Ceyx sangirensis)
12211 Red-throated Lorikeet (Vini amabilis)
14730 Stresemann's Bristlefront (Merulaxis stresemanni); Likely extinct since ca. 2019 in Brazil
15473 Kinglet Calyptura (Calyptura cristata); Until recently, considered extinct, rediscovered in Brazil 1996, subsequently lost again.
18396 Western Wattled Cuckooshrike (Lobotos lobatus); Very few recent records from its SW African range
18397 Eastern Wattled Cuckooshrike (Lobotos oriolinus); Very few recent records from its Central African range
19787 Ua Pou Monarch (Pomarea mira)
21991 Blue-wattled Bulbul (Brachypodius nieuwenhuisii); Very poorly known from its Malaysian range, likely a hybrid or colour variant
22393 Red Sea Cliff Swallow (Petrochelidon perdita); Known only from single type specimen, found dead in 1984 in Sudan
23280 Dusky Tetraka (Xanthomixis tenebrosa); Very poorly understood and apparently rare in its Madagascar range
23373 Tana River Cisticola (Cisticola restrictus); No records since 1972 and only few specimens avalaible, from its Kenya range. Possibly extinct, or may be a hybrid
26183 Bahama Nuthatch (Sitta insularis)
26484 Pohnpei Starling (Aplonis pelzelni)
28435 Spectacled Flowerpecker (Dicaeum dayakorum); Poorly understood with very few sightings in its Borneo range
29718 Shelley's Crimsonwing (Cryptospiza shelleyi); Rare throughout small range in DRCongo, Burundi and Ruanda
B)
109 Tepui Tinamou (Crypturellus ptaritepui)
975 Black Partridge (Melanoperdix niger)
1338 Vietnamese Crested Argus (Rheinardia ocellata)
1339 Malay Crested Argus (Rheinardia nigrescens)
1379 Hainan Peacock-pheasant (Polyplectron katsumatae)
1579 Manipur Bush Quail (Perdicula manipurensis)
2138 Whitehead's Swiftlet (Aerodramus whiteheadi)
2142 Mayr's Swiftlet (Aerodramus orientalis)
2191 Three-toed Swiftlet (Aerodramus papuensis)
3112 Buff-breasted Sabrewing (Campylopterus duidae)
4519 Tooth-billed Pigeon (Didunculus strigirostris)
12559 Louisiade Pitta (Erythropitta meeki)
16915 White-chinned Myzomela (Myzomela albigula)
20662 Snow Mountain Robin (Petroica archboldi)
26456 Mountain Starling (Aplonis santovestris)
31679 Vilcabamba Brushfinch (Atlapetes terborghi)
32461 White-faced Redstart (Myioborus albifacies)
32462 Guaiquinima Redstart (Myioborus cardonai)
32729 Duida Grass Finch (Emberizoides duidae)
33544 Scaled Flowerpiercer (Diglossa duidae)
C)
461 Laysan Duck (Anas laysanensis); Northwestern Hawaiian Islands not accessible to the general public
4426 Socorro Dove (Zenaida graysoni); Revillagigedo Islands not accessible to the general public
5166 Gough Moorhen (Gallinula comeri); Tristan da Cunha Archipelago not accessible to the general public
5242 Inaccessible Island Rail (Laterallus rogersi); Tristan da Cunha Archipelago not accessible to the general public
6572 Ainley's Storm Petrel (Hydrobates cheimomnestes); Guadalupe Island not accessible to the general public
11360 Kakapo (Strigops habroptila); Codfish and Anchor Island not accessible to the general public
11845 Socorro Parakeet (Psittacara brevipes); Socorro Island not accessible to the general public
12157 Night Parrot (Pezoporus occidentalis); Pullen Pullen Reserve not accessible to the general public
22978 Millerbird (Acrocephalus familiaris); Northwestern Hawaiian Islands not accessible to the general public
25912 Clarion Wren (Troglodytes tanneri); Revillagigedo Islands not accessible to the general public
25947 Socorro Wren (Troglodytes sissonii); Revillagigedo Islands not accessible to the general public
26353 Socorro Mockingbird (Mimus graysoni); Revillagigedo Islands not accessible to the general public
27931 Blackthroat (Calliope obscura); Foping and Changqing Nature Reserves not accessible to the general public
30471 Laysan Finch (Telespiza cantans); Northwestern Hawaiian Islands not accessible to the general public
30472 Nihoa Finch (Telespiza ultima); Northwestern Hawaiian Islands not accessible to the general public
31349 Guadalupe Junco (Junco insularis); Guadalupe Island not accessible to the general public
33050 Vampire Ground Finch (Geospiza septentrionalis); Isla Darwin not accessible to the general public
33439 Inaccessible Island Finch (Nesospiza acunhae); Tristan da Cunha Archipelago not accessible to the general public
33446 Gough Finch (Rowettia goughensis); Tristan da Cunha Archipelago not accessible to the general public
D)
3450 Nubian Bustard (Neotis nuba)
4475 Sulu Bleeding-heart (Gallicolumba menagei)
8104 Sulu Boobook (Ninox reyi)
9048 Sulu Hornbill (Anthracoceros montani)
10528 Sulu Pygmy Woodpecker (Yungipicus ramsayi)
11463 Niam-niam Parrot (Poicephalus crassus)
12533 Gurney's Pitta (Hydrornis gurneyi)
18034 Yellow-crested Helmetshrike (Prionops alberti)
18218 Jerdon's Minivet (Pericrocotus albifrons)
18286 Grauer's Cuckooshrike (Ceblepyris graueri)
19708 Bedford's Paradise Flycatcher (Terpsiphone bedfordi)
20250 Hooded Treepie (Crypsirina cucullata)
21415 Burmese Bush Lark (Mirafra microptera)
21429 Ash's Lark (Mirafra ashi)
21441 Obbia Lark (Spizocorys obbiensis)
21703 Prigogine's Greenbul (Chlorocichla prigoginei)
23468 Black-tailed Cisticola (Cisticola melanurus)
23765 Kabobo Apalis (Apalis kaboboensis)
25032 Naung Mung Scimitar Babbler (Napothera danjoui)
25304 White-throated Babbler (Argya gularis)
25312 Chapin's Babbler (Turdoides chapini)
26203 White-browed Nuthatch (Sitta victoriae)
28820 Prigogine's Double-collared Sunbird (Cinnyris prigoginei)
28829 Rockefeller's Sunbird (Cinnyris rockefelleri)
29103 Afghan Snowfinch (Pyrgilauda theresae)
29133 Kordofan Sparrow (Passer cordofanicus)
29318 Lufira Masked Weaver (Ploceus ruweti)
29347 Cinnamon Weaver (Ploceus badius)
29353 Golden-naped Weaver (Ploceus aureonucha)
29401 Yellow-legged Weaver (Ploceus flavipes)
29804 Black-lored Waxbill (Estrilda nigriloris)
31699 Grey-crowned Tanager (Phaenicophilus poliocephalus)
Rejected proposals:
744 Horned Guan (Oreophasis derbianus); Original designation: B?; Reason for rejection: Available in Guatemala and Mexico
1013 Sumatran Partridge (Arborophila sumatrana); Original designation: B; Reason for rejection: Though difficult, available on Sumatra
1182 Siberian Grouse (Falcipennis falcipennis); Original designation: D; Reason for rejection: Visiting Russia may be considered problematic by many, but should pose no imminent danger to ones life for most
1345 Congo Peafowl (Afropavo congensis); Original designation: D; Reason for rejection: Region of occurrence considered rather safe, just very pricey
1464 Whyte's Francolin (Scleroptila whytei); Original designation: A; Reason for rejection: Relatively recent split, available in Zambia
1586 Mount Cameroon Spurfowl (Pternistis camerunensis); Original designation: B; Reason for rejection: Not difficult to hear
1592 Djibouti Spurfowl (Pternistis ochropectus); Original designation: D; Reason for rejection: Djibouti no longer considered dangerous
3108 Santa Marta Sabrewing (Campylopterus phainopeplus); Original designation: A; Reason for rejection: Accessible population found in Colombia
3955 Somali Pigeon (Columba oliviae); Original designation: D; Reason for rejection: Somaliland no longer considered dangerous
4320 Blue-eyed Ground Dove (Columbina cyanopis); Original designation: C; Reason for rejection: Permit currently easily obtainable
6454 Emperor Penguin (Aptenodytes forsteri); Original designation: B; Reason for rejection: Expeditions, though pricy, don't need be organized by oneself
8298 Sandy Scops Owl (Otus icterorhynchus); Original designation: B; Reason for rejection: Though difficult, available in Cameroon and Gabon
8678 Omani Owl (Strix butleri); Original designation: A?; Reason for rejection: Available in Oman (and perhaps elsewhere)
9062 Narcondam Hornbill (Rhyticeros narcondami); Original designation: C; Reason for rejection: May be seen from boat
13861 Alagoas Antwren (Myrmotherula snowi); Original designation: C; Reason for rejection: Permit currently easily obtainable
20277 Iranian Ground Jay (Podoces pleskei); Original designation: D; Reason for rejection: Visiting Iran may be considered problematic by many, but should pose no imminent danger to ones life for most
21196 Lesser Hoopoe-lark (Alaemon hamertoni); Original designation: D; Reason for rejection: Somaliland no longer considered dangerous
21430 Somali Lark (Mirafra somalica); Original designation: D; Reason for rejection: Somaliland no longer considered dangerous
21434 Kordofan Lark (Mirafra cordofanica); Original designation: D; Reason for rejection: Available in northernmost Senegal (and perhaps elsewhere)
23032 Large-billed Reed-Warbler (Acrocephalus orinus); Original designation: A; Reason for rejection: Available in southern Tajikistan
29172 Abd al kuri Sparrow (Passer hemileucus); Original designation: D; Reason for rejection: Visiting Socotra is no longer considered dangerous
29766 Cinderella Waxbill (Glaucestrilda thomensis); Original designation: B; Reason for rejection: Available in Angola and Namibia
30469 Akikiki (Oreomystis bairdi); Original designation: C; Reason for rejection: Still gettable without permit
30491 Akohekohe (Palmeria dolei); Original designation: C; Reason for rejection: Though difficult, may be found outside of restricted areas, access to restricted area possible through organized walks
30508 Maui Parrotbill (Pseudonestor xanthophrys); Original designation: C; Reason for rejection: Though difficult, may be found outside of restricted areas, access to restricted area possible through organized walks
30513 Akekee (Loxops caeruleirostris); Original designation: C; Reason for rejection: Still gettable without permit
30570 Somali Golden-winged Grosbeak (Rhynchostruthus louisae); Original designation: D; Reason for rejection: Somaliland no longer considered dangerous
30695 Warsangli Linnet (Linaria johannis); Original designation: D; Reason for rejection: Somaliland no longer considered dangerous
30631 Yemen Serin (Crithagra menachensis); Original designation: D; Reason for rejection: Available in Saudi-Arabia and Oman
30694 Yemen Linnet (Linaria yemenensis); Original designation: D; Reason for rejection: Available in Saudi-Arabia
32703 Cherry-throated Tanager (Nemosia rourei); Original designation: C; Reason for rejection: Permit currently easily obtainable
33031 Mangrove Finch (Camarhynchus heliobates): Original designation: C; Reason for rejection: Still gettable without permit
This list follows IOC 13.1.
I'm trying to make the designations understandable. Please tell me, if you find errors or if you disagree with my assessments