How can I remove residue left from peel and stick tiles?
The approach is usually progressive. You try a less aggressive cleaner to see if it works, and if not, then move on to a more aggressive. At some point, you may find that the cleaner damages the surface, so test in an inconspicuous area.

I would use paper towels and a mild scrubbing pad, and I would use cleaners in the following order:


a soap based cleaner, such as Murphy's Oil Soap
strong household cleaner with grease cutter, such as Formula 409
a residue remover, such as Goo-Gone
a mild solvent, such as denatured alcohol
a stronger solvent, such as mineral spirits
a mild stripper, such as a citrus based product
a very strong solvent, such as acetone or lacquer thinner (but at this point you are almost sure to damage the surface of the flooring finish).