When I worked there, we were told that the official US Treasury rule is that 51% of a bill is still considered legal currency. Of course, how in the world could you tell the difference between 49% and 51%? You can't, exactly. We did once have some guy come in and try to exchange a few precisely cut bills for new ones. He was smart enough to not include two halves with equal serial numbers, but he was obviously trying to turn one bill into two.
So how did we handle it? Serial numbers. In order to exchange a damaged bill for a new one, we had to have all of one serial number and at least one whole digit of the second. Taped up, stapled or paper-clipped back together from shards was fine (I've seen it all...), but we just had to have those serial numbers. That guaranteed that we were replacing one old bill with one new bill.
In the case of the burnt bag of money, you can imagine what it looked like. As I recall, there weren't but a handful of bills that we could salvage. Hard to say how much was supposedly in there, but I'd guess maybe 5% of the value, based on the fragments we found. Most of it was ash or burnt beyond recognition. I never did learn the story behind it, but it was a good example of why it's not a good idea to keep a lot of money lying around.
And all those old bill remnants get sent to the nearest Federal Reserve for destruction.
The *vast* majority of damaged bills were simply ripped at a corner, which was NBD to replace. We'd occasionally get some that had been through a serious wash though. We treated these with some degree of suspicion, because that's a major way that counterfeiters try to pass their bills. I actually caught a couple of counterfeit bills while I worked there. Always fun to get to call the Secret Service.