I have used cork bar tape on my flat-bar bike for many years without problems. It is a rather old tradition to use (cork) tape on flat bars, gull wing bars, or mustache bars on bicycles.
Here is what I have done: I started with wrapping a used cut up inner tube around the handlebar and tape it down. I am not quite sure if there is an actual dampening effect (my original goal), but it probably works as a anti-slip layer and makes the handle bar a little thicker. Wrap the bar tape tight around the handlebar, and observe that it is wrapped in the correct direction on each side, so that the natural forward pushing hand pressure tightens the bar tape instead of loosening it.
Apply a liberal amount of strong quality tape (Gaffer tape?) on both ends, the supplied tape is not enough. While you can use bar end plugs just like on drop bars to avoid using tape at the end of the handlebar, my experience says that the ends of flat bars gets banged up much more frequently which tend to unwind the bar tape.

The bar ends get wrapped separately, again take notice of the direction and use plenty of tape to secure the ends. My bar end wraps have been exchanged once because of wear and tear, but on the handlebar the tape is perhaps 3 years old and shows no sign of needing an exchange anytime soon. It may look unconventional with cork tape on bar ends but functionality and feel is very good.

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Regards