Time-Turners are in frequent and authorized use by a fair number of students, as well as Dumbledore himself and possibly other staff members. Any such ward would be constantly triggering, for no particularly important reason. Even if one exists, I can't imagine that Dumbledore pays much attention to it.

Dumbledore is only looking for Hermione. There's no reason for him to be much interested in Harry's magical devices, beyond proving that they aren't Hermione. The portkey was only mentioned because it was a part of Dumbledore's body search; the Time-Turner, presumably, wasn't on his body.