Alaska: No trademark may be registered if it disparages, suggests a false connection with national symbols, or bring such national symbols into disrepute.
There doesn't appear to be any laws making it a crime to desecrate the American flag, burn the American flag, or even to use the American flag in advertising. The closes is the above, which prohibits registering trademarks which disparage national symbols like the American flag.
- Overview of State Laws on Flag Abuse, Burning, Desecration
- Index of State Laws on Flag Abuse, Burning, Desecration