Sometimes those with whom we do not agree bring to light other aspects of an issue that may need examination. So protestors of all kinds should absolutely be protected by the First Amendment.
Now, burning someone in effigy seems like a cathartic release, for some people. If there is an especially tryannical or crass despot or an inexperienced, unknowledgeable and dishonest "leader," the burning might relieve some emotional pressure for some of the citizenry.
Suggestions have been made for an effigy of the congressman who asked a Naval officer if Guam was going to tip over when there were too many people on one side of the island. (true! His very words were on the news and his inane video is on Utube.)
Since freedom of speech is one of our constitutional rights, we must protect it. As burning an effigy is symbolic speech, it must be allowed. Regardless of the speech, whether we like it or not, we must accept it. The things that would prohibit this would be public safety and possibly laws that prohibit certain types such as threats against the president.
We have to allow the First Amendment to protect all freedoms of speech, even those that we disagree with or that are unpopular. That is the only way we can say we have the protection of freedom of speech. A recent example is the church which goes around protesting at soldiers funerals, I strongly disagree with them and wish they would not protest but I also realize we have to allow them that freedom.
Perhaps the best answer to this question was the opinion of Justice Brennan in Brown vs. Texas, which involved burning the American flag. Justice Brennan said that burning the flag was one of the rights which the flag itself stood for. He also suggested that the best response when one burns the flag is to wave the flag. i didn't often agree with Justice Brennan, but I had to that time.
A more pointed example would be Phelps vs. Snyder which the present court heard in its most recent term. ("Thank God for dead soldiers, etc.") The court held, correctly, that the speech protected by the First Amendment is unpopular speech. Burning an effigy is a form of symbolic speech, and is the very type of speech that the First Amendment is designed to protect. All of us, at one time or another, are in the minority. We should all be thankful that we have the right to disagree.
As I've gotten older and employed and found that you cannot speak about what happens at work (even if you work in a place like a "public school,") I've gained a greater appreciation for the first amendment.
And I completely agree with Pohnpei, as long as burning something or someone in effigy doesn't create harmful situations or is done in an inappropriate place or time, I think the first amendment should absolutely protect the right of someone to do that. I don't know that I personally would ever go to that extreme, but that doesn't matter. People's right to expression is super important and protecting it is vital.
It does and it should.
Burning people in effigy is a form of speech, something called symbolic expression. It is a way for people to express ideas in ways that are not verbal. Symbolic speech should only be banned if the acts involved actually hurt people. So, for example, you should not be allowed to express your ideas by burning someone's house because that would cause them harm.
But burning an effigy does not do a person any real harm. It does express deep disdain or contempt for the person, but it does not physically harm them.
So, as long as the burning is not done in an unsafe manner and as long as it is not done at the wrong time or place (on the person's lawn, perhaps, or in the middle of the street at rush hour) burning effigies is and should be protected by the First Amendment.