The general idea is that host cell line infections can occur (TVT, Tasmanian devil facial tumor, and a contagious leukemia in Syrian hamsters), can mutate into something that is nonlethal and/or chronic if selection favors that (TVT usually goes away with time), can infect related species (TVT can be experimentally transmitted to wolves, jackals, coyotes and red foxes), and might exist in humans today. There are human diseases that appear infectious for which the transmissible agent has not been identified - sarcoidosis, for example. So modern humans might suffer from infectious organisms directly derived from Neanderthals or other archaic humans. As far as I know, no one has yet thought of looking for Neanderthal-derived cells inside people. Since such cells would have the required genetic code for making human signal molecules, they might be particularly likely to employ baroque forms of host manipulation.
… there _could_ be Neanderthal-derived cell line infections, and this is really the only scenario I've been able to come up with that gives us live Neanderthals - hiding in your sinuses, or maybe your prostate. The only one so far. There are other known infectious diseases in which some metazoan has completely chucked complexity and gone back to being a germ: whirling disease in fish, for example.
Will it then be possible to resurrect Neanderthals à la Jurassic Park? Another GNXP commenter, Eric J. Johnson, poured cold water on the idea:
… the problem would be a lack of purifying selection on all the morphogen genes, not to mention all the neuron-specific genes, etc. The tumor doesn't need any of that stuff. How fast they would all turn to garbage, I don't know. Probably pretty fast.