I just had to answer this. Too funny. No, it won't hurt your chickens' nails to paint them. Be very sure they're completely dry, though, before you turn them loose to model their new look for the flock.
Because all the others will zero in on those fantastic colored nails and will likely peck the daylights out of them, probably including the one sporting the nails. If the paint isn't totally dry, the thinner in the base that makes it liquid so you can paint it is toxic and can cause brain damage. But it's harmless when dry.
Tip: choose neutral colors like green and blue and purple. Reds and oranges excite chickens and they're more likely to want to peck at those colors than the cooler shades.
Now, we are all expecting photos of those flamboyant talons! Please don't disappoint!