What is the best way to remove peel and stick vinyl tiles?

Try heating them first.
Go down to a re-sale shop and find an old iron you won't feel guilty wrecking.
Then find a really wide putty knife, the kind they use to tape joints in wallboard.
Set the iron at the lowest setting and heat up the tile, use the putty knife to peel it up.
Bring the temperature of the iron up a little at a time until you find just the right temperature to soften the glue with out cooking the tiles. You may have to peel off the vynal coating and apply the heat to the paper backing to get the glue to soften.
This is a long and ugly process but the only one I know of.