It takes just under 90 seconds for the money to burn. This time appears to be the same across difficulties and Regular/Pro. As you can see in the screenshots below (taken immediately after the money was ignited and immediately after the camera became available for pickup on hard and overkill difficulties respectively) the duration is about 88 to 89 seconds in both cases.
Start (Hard Difficulty) - 9:21
End (Hard Difficulty) - 10:49
Start (Overkill Difficulty) - 7:07
End (Overkill Difficulty) - 8:36
It's possible that (like so many other things in Payday 2) the duration is based on a random number generator and I just happened to have two of the same durations, but for now I'd say that the duration is most likely just under 90 seconds.
Update: I just timed the burning duration on Overkill (again) as well as Pro Hard, and both were almost exactly 90 seconds.