The Supreme Court most recently answered the question in 1989. It's decision, in Texas v. Johnson, the court affirmed a lower court decision dismissing the claim against the flag burner.
The court said:
The way to preserve the flag's special role is not to punish those who feel differently about these matters. It is to persuade them that they are wrong. "To courageous, self-reliant men, with confidence in the power of free and fearless reasoning applied through the processes of popular government, no danger flowing from speech can be deemed clear and present, unless the incidence of the evil apprehended is so imminent that it may befall before there is opportunity for full discussion. If there be time to expose through discussion the falsehood and fallacies, to avert the evil by the processes of education, the remedy to be applied is more speech, not enforced silence." Whitney v. California, 274 U.S. 357, 377 (1927) (Brandeis, J., concurring). And, precisely because it is our flag that is involved, one's response to the flag [491 U.S. 397, 420] burner may exploit the uniquely persuasive power of the flag itself. We can imagine no more appropriate response to burning a flag than waving one's own, no better way to counter a flag burner's message than by saluting the flag that burns, no surer means of preserving the dignity even of the flag that burned than by - as one witness here did - according its remains a respectful burial. We do not consecrate the flag by punishing its desecration, for in doing so we dilute the freedom that this cherished emblem represents.