Lack of tape can be a major hazard (well, to be honest, mostly it's a major hazard if you are playing with the spoke tension) - definitely should go right up the FIX IT NOW list, plus it's going to screw up the inner tube. Bike shop are completely thick to have sold you an inner tube of the right size with rim tape of the wrong length!

Gaffer tape should not be used as a replacement for rim tape, however putting gaffer tape around the innertube is plausible (but means taking the damn tire off again).