Let me quickly deal with the second (minor) part of your question before we jump into the main one:
Is there any political group of Muslims who advocate the right to do things because they believe that non-Muslims are not bound to the religious expectations of practising Muslims, etc.?
Since the time of the Islamic era, when Islamic empires ruled over vast majority of non-muslims too, to modern Islamic nations today, muslims have never subscribed to the belief that non-muslims should be bound to the religious practices of Islam. This is easily apparent when you examine how non-muslims, their religion and their culture have thrived (and continue to thrive) in Islamic nations. For example, idolatry is against the tenets of Islam (and a very foundational and strong tenet), and yet, Churches and Temples are still built in many middle-east Islamic nations just as they were in the Islamic empires. Pork is forbidden to muslims and yet non-muslims have been raising pigs for food centuries even during the Islamic era, and their consumption was not prohibited to the non-muslims, nor were they treated differently for consuming it.
(The reason why people errantly and ignorantly believe that muslims like to impose their religious beliefs on others comes from the imposition of Sharia laws on non-muslims in some Islamic nations. However Sharia is simply a legal system based on Islamic values, and not an imposition of the religion itself. Like any legal systems, it has its pros and cons, but often, people only focus on the cons.)
Coming back to your main question:
Is there any political group of Muslims who advocate the right to do things like burn the Quran, ie, because they believe in free speech?
This is a loaded question about:
- Free speech in a democracy.
- Politics of religion ("Where are the moderate muslims and why aren't they speaking up?").
Your question presumes that "free speech" only means the absolute right to say anything on any subject. But is that the reality of the democratic world? Even if you subscribe to the belief that only the western nations are truly democratic, you'll find that even they can't agree on how much it should be regulated. Here's a good example of it, related to both the subject (Islam and Free Speech) we are talking about:
In the fall of 2009, a woman referred to as E.S., a 47-year-old Austrian national, convened two seminars offering “Basic Information on Islam” at the right-wing Freedom Party Education Institute (FPEI) in Vienna ... an undercover journalist ... lodged a complaint against E.S. to the police. The substance of the complaint related to two comments E.S. made in the course of a discussion about Muhammad’s marriage to Aisha. He “liked to do it with children,” she asserted, adding, “A 56-year-old and a six-year-old? What do we call it, if it is not pedophilia?”
The ECHR ruling came on appeal. On February 15, 2011, the Vienna Regional Criminal Court found E.S. guilty of publicly disparaging religious doctrines, which is a crime in Austria ... In its judgment, it affirmed that “freedom of expression constitutes one of the essential foundations of a democratic society.” It also acknowledged that religious minorities “cannot expect to be exempt from criticism” and “must tolerate and accept the denial by others of their religious beliefs and even the propagation by others of doctrines hostile to their faith.” However, the ECHR was adamant that “the exercise of the freedom of expression carries with it duties and responsibilities, including “the duty to avoid as far as possible an expression that is, in regard to objects of veneration, gratuitously offensive to others and profane.” It also made clear that where criticism of religion takes the form of “an improper or even abusive attack on an object of religious veneration” it should be considered as exceeding the bounds of acceptable expression, since it is “likely to incite religious intolerance.” - A Flawed European ruling on Free Speech
(Note: I agree with the ruling and find it logical and sensible but have still cited an article critical of it to showcase the ongoing interesting debate on free speech.)
I bring up this issue about difference in opinion about free speech in a democracy to highlight an important point - as the chart above conveys, the vast majority of muslims in the world are in Africa, Eurasia, the middle-east and Asia. And all the country in these regions, even the most democratic ones, regulate speech much more strongly than the west. Thus, culturally, the majority of muslims don't subscribe to the "western" notion of free speech.
There's a good article I came across - Free Speech and the Muslims Who Love It - and one point in it struck me:
The main problem on both sides has been an emotional response taking hold before a logical one ... People (not just Muslims) who feel disempowered often resort to emotional displays as a release, with no reasonable expectation that the source of the provocation will change. The desire to do something – anything – to address a wounded soul comes first and foremost.
This is the crux of the matter - free speech expects rationality in everyone. And this is unrealistic when there is no political "release valve" for pent-up frustration.
A non-muslim may a burn a Quran out of frustration against the "intolerant" muslim(s). This frustrates some vulnerable muslims. Some of these muslims react emotionally with further hate and violence. And the cycle continues.
The real issue is not free speech at all - the actual issue is whether there is a political outlet to express it safely, rationally before it becomes a pent-up emotional mess that one feels can only be expressed dysfunctionally.