Use WD40 or Xlite's Bike Spray. Honest, it works better than anything else. Spray it on and give it a good rub with a clean cloth. My white bar tape now looks as good as the day I fitted it even after a year and a half of commuting.
Should be fine. I don't get on with that shiny plastic coated bar tape so I'm using Specialized Bar Phat cork tape at the moment and have had 3TTT cork tape in the past and have had no problems cleaning either using the above method.
I should point out that WD40 was what I was originally advised by a few friends but I've just realised I've only ever used Bike Spray. It's an almost identical product but if you use WD40 give it a shot on a small bit first just in case. I wouldn't want to be responsible for recommending something that detroys your tape.