It depends on the type of stopper. And, yes, some hair caught could cause capillary action and drain the tub, but that would normally be quite slow and unlikely to hear it.
Is the stopper all internal, or can you see something move up and down in the tub other than the lever? If it is all internal, it may be a sleeve that gets moved up and down with the lever. It often has an o-ring on it, and changing the height aligns a hole in it with the drain. Adjust it, and it may always have an opening.