Every time I grease my TW drive shafts I do my best to remove the excess that oozes out after pumping in the new stuff. But some always get flung out onto the exhaust during the first drive - and makes a burning smell for a couple of days... Can this stuff catch fire?

 

It does have a flash point. I believe it's around 400 degree F. Temps at the manifold can definitely exceed that. Hard to say how fast it dissipates.

Haven't heard of someone having an issue, but I can't confirm either way.

 

The issue/concern is when the oil will separate and when that will catch fire. Not all greases are the same, did a test a couple of months ago with a torch, a heat gun and some different greases looking for a good high temp solution. Most started separating somewhere between 295° and 350° while a couple did earlier and one held in place without sliding off or separating until almost 550°. The winner isn't a grease they sell off the shelf though, and they were only willing to sell it in 55 gallon drums for a whole lot of money $3,295.00 for a 55 gallon barrel. Great stuff, but not on my budget. Heck, I probably couldn't use that much in my lifetime anyway.

The fire part.... They all caught fire and burned once the oil separated if temp was above around 450°

 

Good chance you're over lubing. Wouldn't want mine on that mountain of burned out jeeps in the scrap yard.
In other words less grease is required and don't lose sleep over grease burning in the first place.