Treason
Whoever, owing allegiance to the United States, levies war against them or adheres to their enemies, giving them aid and comfort within the United States or elsewhere, is guilty of treason and shall suffer death, or shall be imprisoned not less than five years and fined under this title but not less than $10,000; and shall be incapable of holding any office under the United States.
While an argument might be made for flag burning being "aid and comfort", I think any prosecutor who tried might have the judge channeling Inigo Montoya. Also, death, or even a minimum 5 years/$10,000 seems a bit harsh; YMMV
Should it be illegal? I say no.
Though I find such activity to be despicible, the flag is -- while highly revered -- merely an icon. It represents the government, the country, both, or something else altogether, depending upon who you are.
It is not, however, a person. It is also not (unless stolen) the private property of another.
Its destruction, therefore, violates no one's rights.
Rights of any kind, by the way, do not owe their existence to any form of thuggery; they were long before any government came into being, and will continue even after they have been outlawed.
Whether you believe in creation or natural law, the concept of rights demand that they exist regardless of the powers that be or the absence of the same. Anything less renders them mere privileges.
But I digress...as I often do.
As I was saying, no harm is done to anyone's rights when a flag is burned. Some people get upset...very upset....and I'm not saying they shouldn't.
However, if flag desecration were made illegal, on what justification would it be so?
It is the offensiveness of the act, which is a representation of thought (i.e. form of speech), that triggers the emotion and lights the desire to see someone punished.
If flag desecration is made illegal in and of itself then a thoughtcrime has been created.
When such laws are made, society tells the actor "You have not harmed anyone, nor have you harmed anyone's property, and therefore have not violated any rights. You have violated sensibilities, you are a criminal because of your thoughts as expressed through your actions."
Who decides what is venerable enough for legal protection against such thought crime?
The federal government only?
Maybe the states?
Maybe Alabama could pass a law against desecration of the Stars and Bars.
Maybe, instead, the waving of certain flags could be outlawed; I'm sure there are some locales that would love to eliminate the same Confederate battle flag, or maybe rainbows, or Lord knows what else that a large enough group (laws don't take a majority, just enough pressure on and fear in the legislature) could push for and win.
Thankfully, while flag burning bills and even Constitutional Amendments have, from time to time, been proposed at various levels, in most cases the governing bodies have, as a general rule, recognized the slippery slope below.
Big Brother help us when they lose sight of that; you can rest assured he will.