Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.
Where in there does it say burning a flag is speech?
Going out chanting and carrying around signs is speech, but I wouldn't consider burning the flag.
Of course one way to settle it is to make it illegal to start a fire in the city limits.