are Muslims permitted to interpret the Qur'an as they wish? Or must it be viewed, literally, in its entirety?
Speaking as an anti-theist and advocate for the end of Islam as a doctrine with religious intent, I currently feel that an accurate answer would be twofold.
On the one hand, I am so often told of the importance of the Qur'an and of how disastrous it is to even take the Ahadith too seriously that it certainly seems difficult to deny that the Qur'an must indeed be viewed (according to Islam) by a perspective that is, if not rigid and literal, at least remarkably careful to choose among often very disparate readings.
On the other hand, that is only possible because the text is not particularly clear to begin with, and therefore it must follow that a good Muslim needs in fact to interpret the Qur'an even if he would rather not. The only alternative would be to let someone else do the interpretation and borrow from there, but I am also told that Muslims don't really approve of that.
A more pragmatic yet perhaps more significant consideration is that most Muslims, like most other people, tend to avoid potentially difficult discussions of proper doctrine unless given either some form of reassurance that those difficult matters will not be raised or else a clear reason to run that risk.
TLDR: No, they are not permitted to freely interpret the Qur'an, mainly because much of the point of having it is to establish clear, rarely questioned paramenters for social behavior.
Yet, at the same time, most if not all Muslims can't help but ultimately accept the need to choose some interpretation among several. They usually won't like to admit that need, but they will feel it nonetheless.