People do actually burn American flags on a semi-regular basis, don't they? I've heard of some court cases stemming from it (which the flag-burners have generally won), but no violence, at least not in the past few decades.

Given that some groups of Muslims have had their places of worship firebombed for the terrible crime of worshiping while Muslim, I have great confidence that if Muslims ever got together in the US for a flag or Bible burning, they would face retaliatory violence.

Yes the worst 'offenders' are the Boy Scouts, Army, Veterans associations and Fire Depts.

See USCODE Section 175(k), Chapter 1, Title 4

Right, so is the main problem with the burning act itself, or with the reaction to it?

When you are performing an act that you know will cause a certain reaction you are either: 1) trying to make a statement or 2) intentionally trying to cause that reaction.