I have absolutely no evidence to support this totally speculative hypothesis, but I'll throw it out anyway.

Burma Shave is not just plain old soap in a can. It's scented. So what does it smell like? Well, some folks with vivid olfactory imaginations may have thought it smelled a bit spicey. So where do exotic spices come from? Asia. So, if the folks that made the scented soap in a can wanted to give it a name to personify its spicey fragrance, it's logical that they would choose the name of an exotic Asian local. "India Shave" doesn't work. How about "Laos Shave"? Nah! "Japan Shave"? Nope. "Burma Shave"? Yeah! That'll work.