Does the lifting of the tax change the demand for gasoline? No. Does it change the physical supply? No. At least not by much in the short run and especially not by much when the tax relief is known to be temporary. Since neither supply nor demand change neither does the price. What does change is that with the tax the government collects the revenues, with the "tax relief" the suppliers of gasoline collect the (former) revenues. Either way, no gain to the consumer.
I agree with Alex if he's talking about the world tax rate. But there is no world tax rate. If one part of the world cuts taxes and the rest doesn't, then gas flows into the lower-tax area, and consumers in that area benefit. This is clearly true for state-level gas taxes. If Virginia dropped it's gas tax, more gas would flow into Virginia. And since the U.S. is just one corner of the world economy, cutting gas taxes in the U.S. would cut U.S. gas prices - even if global production is unchanged.
I'm not saying, of course, that the man-in-the-street understands any of this. But his knee-jerk conclusion is correct. The worst you can say about it is that it's incomplete: At least in the short-run, cutting gas taxes reduces your gas prices by raising them everywhere else.