My experience with this is that the are VERY hard to remove completely. They crack, split, and more often than not the glue stays on the floor and the tile comes up in pieces, leaving a bigger mess than when you started. I just put laminate right over them and it worked fine. I'm not sure about hardwood, but I don't see why it wouldn't work fine also.
if you nail down hardwwod over the tiles, they could loosen and/or break up leaving voids which in turn could cause your floor to develop squeaking over time. Many floors have been installed over old va tile to encapsulate and have been done so successfully but many have been unsuccessfull as well. its a hit or miss situation that could simply be avoided by removing the old tiles.
Good Luck in whatever decision you make.