In the old days no bar tape had adhesive, and it was fine.
The adhesive is suposed to help the wrapping process, once wrapped properly I haven't heard anything to indicate that it enhances the durability or resistance to unwrapping...
You'll probably be OK. Just be sure to maintain a good amount of tension on the tape as you wrap it.
One trick that I sometimes used to do was to loosely pre-wrap the bar with friction tape (not the same as plastic clectrical tape). That provided some stick-um to help hole the handlebar wrap in place.