Risks of latent viruses that reside in ancient genomes under research?
First, I don't understand why you are more worried about viruses of extinct species, instead of ancient viruses of species that haven't gone extinct. Clearly the infectious potential of the latter is greater for the current species. I haven't heard a peep in published literature about the dangers of resurrecting mammoth viruses, but surely recreating (in 2005) the 1918 influenza
did stir some fairly substantial safety debates.

In general, working with ancient DNA (aDNA) does require some safety precautions, but mainly because of the risk of contaminating the aDNA (with modern DNA)-.

Furthermore, I'm not aware of any research having resurrected an ancient virus from an extinct species aDNA (your specific scenario), but one ~5-million y.o. retrovirus (dubbed Phoenix) was assembled from current human DNA (in 2006); the authors were a little worried as to its infectious potential, so they engineered it so it couldn't replicate more than once.

More commonly, the genome of less ancient virus strains is being fairly routinely reassembled from e.g. from mummies, skeletons, or even packrat middens in order to study the evolution of common viral diseases. Most of this goes on without any safety controversies, as far as I can tell. Less commonly, ancient viruses of various species are found fairly intact in permafrost which does also stir some safety discussions (regarding the handling of permafrost).

So is your scenario possible in theory? Yes. Are experts worried about it? Not that I've heard...