Should loose underlayment nails be driven in or removed?

Are they ring shank nails? If so you should be ok hitting them back in. If not, I would remove and use deck screws to fasten the subfloor back down.

If they worked their way out now (due to shrinkage in lumber, nails too short, deflection of joists, etc.) they could work their way out again.

I’d remove them, fill in the holes and re-nail or screw the underlayment down.

Depends how thin the vinyl is. They probably are proud due to wood shrinkage over time/humidity etc. Removing them may not even be followed by filling holes - depends how wide they are. Nails used for floorboarding are usually lost head, so hammering them level would be all that's needed.

If you wanted, you could remove the proud ones and fill, or drift them just below surface and fill, but with decent thickness vinyl there should be no need.