Yes, and it is happening as we speak.
Many tropical reserves are now living on borrowed time. They are in debt after 2 years of no tourists due to the Covid restrictions, and cannot survive much longer. Now it increasingly looks that tourists will not come for another season.
Possibly the only exceptions are countries like Kenya which are highly dependent on tourism and where general tourist interest is so high that there's a large and constant flow.
Far from 'exceptions' , foreign tourism kept alive all savanna animals in East Africa, most of South Africa, reserves for mountain gorillas, Galapagos, Costa Rica, Madagascar, and a large number of smaller reserves created for single endemic sites. From top of my head.
Local ornithological societies in the tropics are usually too small to significantly support conservation (and many of their members indirectly benefit from tourism, too). By the time they would grow to the level of, say, Birdlife in Britain or ABA, most habitat and species in the tropics will be already gone. Local tourism rarely generates similar funds, often there are day visits or sponsored visits like schoolchildren tours which don't generate much money. Or rich locals are interested in a standard nature damaging tourism (a party hotel instead of a rainforest trek).
If tourists don't come, much of biodiversity will be lost long before (if) climate change would finish them. And the loss of forests etc. in these reserves will probably faster the climate change anyway. By the way, the biggest threat may not be the ideological singling out long-distance aviation as a supposedly unneeded CO2 producer, but the increasing poverty in the middle class in Western countries.
BTW, another thing will be ripple effects of reduced travel on domestic birdwatching. Many reserves in Europe or the USA also significantly depend on money and support of tourists from far parts of the same countries. And if all traveling birders and twitchers are gone, the local birdlife societies will have significantly less money. And the remaining members will have reduced interest in protecting wildlife in far-off countries they will never see. They will see their duty to protect nature as creating a local reserve for their local suburban birds, not the exotic rainforests or savannas. Out of sight, out of mind.