What happens if I put my foot in white vinegar for 10 minutes or longer?

It's cleansing. No harm done. White vinegar diluted (1:4) is a home remedy for many things in the foot besides smelling a little "pickled", the vinegar is actually antimicrobial (great for minor cuts and abrasions), antifungal (great for athlete's foot) and is a keratolytic (great for calluses).