Hitler was definitely shot to death by the Basterds but he would have died in the fire anyways. Shoshanna had everything planned well to trap everyone in theatre including her, there was no escape. So when looking at this movie if the parts with the Basterds were removed her plan would have still worked. So were they even needed?
Recall how Marcelle does the final preparations for setting the theater on fire - he blocks the auditorium doors with a crowbar and then proceeds to behind the screen and starts waiting for Shoshanna's "message" to appear on the screen. Before he does so the two men from the Basterds leave the auditorium and proceed to some side corridors to get to the side balcony and attack Hitler. Some time after they leave the auditorium Marcelle comes to the doors and blocks them. The balcony where Hitler sits with Himmler has a separate door guarded by two German soldiers. The two Basterds kill the guards, rush into the balcony and shoot Hitler. Right after that the "message" appears and Marcelle sets the pile of films behind the screen on fire.
So the balcony doors are not blocked - Hitler likely can escape the theater or at least gain some time and space and have better chances to have been resqued. He definitely has much better chances than those people who are in the auditorium - those are blocked between the locked doors and bursting flames - but he is at least not in the auditorium and so less exposed to the dangers and has a chance to use the side corridors to escape. The two Basterds shooting him on the balcony make that totally impossible.
My guess is that more people died than would have otherwise: the survivors not only needed to avoid the fire; they needed also avoid the bullets. That is, I don't expect that everyone would have been killed by the fire, nor by both. Maybe Hitler would have even survived it, given the level of devotion his minions possess... someone would have sacrificed self to save him, for example. Bullets are a lot harder to dodge than fire :)
When the Basterds were first unleashed by the Allies, they had absolutely no idea that someone else was working on reaching the same goal. They didn't know about Shosanna. In fact, it was only because of Zoller's infatuation with Shosanna that her small theater was even chosen for the premiere of Nation's Pride.