As I said in this question, time turners are very simple: just turn the hourglass, and poof! You're back in time. But does it require magic? Could a Muggle do this?
The entry on pottermore regarding time-turners states that the charm needs to be worn around the neck of a witch or wizard. While this doesn't preclude their use by other beings (magical or otherwise), it certainly isn't the most inclusive of statements either.
"As our investigations currently stand, the longest period that may be relived without the possibility of serious harm to the traveller or to time itself is around five hours. We have been able to encase single Hour-Reversal Charms, which are unstable and benefit from containment, in small, enchanted hour-glasses that may be worn around a witch or wizard's neck and revolved according to the number of hours the user wishes to relive."
I have no canon answer, but I think that a Muggle could use it. A Time Turner is an object which can transport you back and forth in time, and it has magic on its own.
The way I see it,it is similar to a Portkey, which can transport you back and forth in space and can be (even accidentally) used by a Muggle, as we can read on Pottermore (see http://pottermore.wikia.com/wiki/Portkeys and https://scifi.stackexchange.com/a/106433/60559). Now, I don't know if time-travelling and space-travelling are very different from the magical standpoint, but it seems to me that, if a Muggle can use a Portkey, he/she can also use a Time-Turner.
In addition to that, we know that Squibs - and probably even Muggles - can enter the Room of Requirement(Could a Muggle Access the Room of Requirement?), which is another object possessing magical properties of its own.