Contrary to the other responses I believe they did not based on Hagrids remark that they might have been able to apply for time turners to take his class in their 6th years, to which they respond they were destroyed at the ministry last year. This would imply that teachers other than heads of house and Dumbledore are aware of it. Not to mention they talk to each other about their classes and students obviously and the fact that she is in multiple classes at once would come up.

Probably not.

Obviously, from Rahul's answer, we know that Dumbledore knew. However, there was no reason other teachers had to know about the Time-Turner. For all they knew, Hermione just walked in after her two friends and there was no reason to be concerned by a simple, "When did she get here?" Of course, we have no indication one way or the other, but I don't believe it would be necessary for the other teachers to know.

DUMBLEDORE

Mysterious thing, time. Powerful. And, when meddled with... dangerous. Sirius Black is in the topmost cell of the Dark Tower. (to Hermione) You know the laws, Miss Granger. You must not be seen. And you would, I feel, do well to return before this last chime. If not... well, the consequences are really too ghastly to discuss. Three turns should do it, I think. If you succeed, more than one innocent life may be spared tonight.