The Danish publication of satirical cartoons of Muhammad generated a lot of heated discussion about the moral and political legitimacy of satirizing or mocking religion, but this issue has generated heated debate for a long time. Muslims weren't the first to seek censorship of images or words which offended them, and they won't be the last. Religions may change, but the basic arguments remain fairly constant and this allows us to more quickly respond when the issue arises again (and again).
Freedom of Speech vs. Morality
There are two fundamental questions at stake in these debates: whether the publishing of offending material is legal (is it protected as free speech, or can it be censored?) and whether it is moral (is it a morally legitimate expression or is it an immoral attack on others?). In the West, at least, it's a settled matter of law that mocking religion is protected as free speech and that free speech rights cannot be limited to just material which no one objects to. Thus no matter how immoral the speech arguably is, it's still legally protected. Even at the fringes where the immorality consists of causing harm, this doesn't always justify restricting speech.
The real debate is twofold: is it immoral to mock or satirize religion and, if this is the case, would this constitute a reason to change the laws and censor such material? The moral question is the most fundamental one and thus the question which must be engaged most directly because if religious believers cannot make the case that mocking religion, religious beliefs, religious institutions, or religious figures is immoral, then there is no reason to even start discussing whether it should be made illegal. Making the case that mockery is immoral is not by itself sufficient to justify censorship, of course, but it is necessary if censorship is ever to be justified.
Mocking Religion Stereotypes Believers & Promotes Bigotry
If successful, this would be the strongest objection to mocking religion. There would still be arguments against censoring such material, but it's hard to argue that it's moral to promote stereotypes of all adherents of a single religion or to promote bigotry against those adherents. This argument is very context-specific, however, because there is nothing about mockery or satire which necessarily leads to stereotypes and bigotry.
Thus religious apologists must establish in ever individual case how a specific example of mockery leads to stereotypes and bigotry. Moreover, anyone making this argument would have to explain how satire of religious beliefs leads to immoral stereotypes while satire of political beliefs does not lead to immoral stereotypes.
Mocking Religion is Immoral Because it Violates Religious Dogma
Most religions have at least an unstated prohibition against mocking revered leaders, scriptures, dogmas, etc., but it's also common to have explicit prohibitions against such expression. From the perspective of that religion, it is mockery and satire would be immoral, but even if we allow that this perspective is legitimate we have no reason to suppose that it must be accepted by outsiders.
It might be immoral for a Christian to mock Jesus, but it can not be immoral for a non-Christain to mock Jesus any more than it is immoral for a non-Christian to take God's name in vain or deny that Jesus is the only means to salvation. It wouldn't be legitimate for the state to force people submit to such religious rules — not even if they are adherents of the religion in question and certainly not if they are outsiders.
Mocking Religion is Immoral Because Offending People is Immoral
Giving offense isn't the same league as lying or stealing, but most people will agree that there is at least something morally questionable about offending other human beings. Since mocking religion can reasonably be expected to give offense to believers, isn't it immoral? Accepting this principle entails treating as immoral anything which might be expected to offend someone, and is there anything that won't offend some hypersensitive person out there? Moreover, if reacting with offense is claimed to be offensive to those doing the original mocking, we would be caught in an endless loop of censorship and accusations of immorality.
Giving offense may be morally questionable, but it can't be immoral enough to demand that the state forcibly stop it. No one has a right to never encounter anything which might offend them. Most people probably recognize this, which is why we don't see calls to punish those who say something offensive in the context of politics.
Mocking Religion is Immoral Because Gratuitously Offending People is Immoral
Mabye we can preserve the argument that offending people is immoral if we set aside the most hypersensitive observers and simply argue that it's immoral when it doesn't serve any legitimate purpose — when we can reasonably expect people to take offense and the legitimate goals we had could have been achieved just as well through non-offensive means.
Who gets to define what qualifies as a "legitimate purpose" though, and thus when offense has been given gratuitously? If we allow the offended religious believers to do it, we will quickly be back where we were in the previous argument; if we let those doing the mocking decide, it's unlikely they will decide against themselves. There is a legitimate argument in saying "don't gratuitously offend," but it's not an argument that can easily lead to accusations of immorality, never mind justify censorship.
Mocking Religion in Particular is Immoral Because Religion is Special
An even less convincing effort defend the argument that offending people is immoral is to say that there is something special about religion. It is claimed that offending people on the basis of religious beliefs is much worse than offending people on the basis of political or philosophical beliefs. No argument is given on behalf of such a position, though, aside from the fact that religious beliefs are very important to people. Furthermore, it's not clear that this escapes any of the circularity problems described above.
Finally, it's not credible that beliefs can be separated so neatly because religious beliefs are also very often political beliefs — for example, when it comes to issues like abortion and homosexuality. If I'm harshly critical of Christian or Muslim positions on gay rights and this offends someone, should this be treated as giving offense in the context of religion or in the context of politics? That matters a lot if the former is subject to censorship but the latter is not.
Mocking Religion is Immoral Because it Leads to Violence
The most curious argument is based on the reactions of people who are offended: when offense is so great that it leads to riots, property destruction, and even death, then religious apologists blame those who published the offending material. It is usually immoral to engage in riots and certainly murder, and it is also immoral to incite riots that lead to murder. It's not clear, though, that publishing offensive material is the same as directly inciting the violence of offended believers.
Can we take seriously the argument that "your satirical material is immoral because it offends me so much that I'm going to go out and riot"? Even if this argument were made by a third party, we're facing a situation where any material will be deemed immoral so long as someone is mad enough to harm others over it. The end result would be a tyranny of whichever special interest group is willing to be violent enough.