How to clean white handlebar tape?
Grab some of the disinfecting wipes in the round containers that you normally use for kitchen counters, etc. I like the Clorox brand in the yellow (Lemon Fresh!). Wrap around bars and squeeze a bit as you twist/rotate in the same direction as the wrap.

With cork based wrap this is fairly effective. With some of the slicker surfaced wraps (Lizard, etc), it's downright awesome. If you start cleaning with the wipes on a regular schedule your wrap will get even cleaner over time.