Does dyeing white hair cause more white hair?

No, not according to any current (as of Fall 2012) scientific research.

While the chemicals contained in hair dye products vary, most boxed home hair color kits come with warnings regarding possible allergic reactions. For this reason, most, if not all, home hair dye kits contain explicit instructions to do a skin allergy test prior to dyeing one's hair.

If someone doesn't properly follow these instructions, they may experience an allergic reaction.

The internal chemicals that cause hair to lose its coloration (because a white or "gray" hair isn't actually white or gray, but translucent, having lost the melanin that makes up the hair's color) could possibly be affected by allergic reactions or other hormone imbalances, but there is no one reason why white or gray hairs appear, regardless of whether you dye, cut, or pluck those hairs.