RAII style API wrapper for PyMongo
Great that you provided a docstring! You made one minor formatting mistake though, there should be a blank line between your brief single line summary and the rest of the docstring. It's recommended in the style guide partially for readability and partially for script parsers.

class MongoDB(object):
    """Provides a RAII wrapper for PyMongo db connections.

    Available collection functions limited to those in
    attributes_to_pass. Number of simultaneous connection 
    users also tracked. """


It seems like attributes_to_pass is actually a constant. If that is the case it should be UPPER_SNAKE_CASE to be clear that it is, especially when mixed with attributes that often change like client and num_users. It should also be a tuple, as tuples are immutable so will not change unless updated in the source code.

ATTRIBUTES_TO_PASS = ("update", "insert", "count")


Also, when you call on these constants you refer to the class name. While that's possible I'd personally say it's less readable. I had thought you were calling on a class you imported for a second. You can just pass self instead of the classname.

def __getattr__(self, attr):
    if attr in self.ATTRIBUTES_TO_PASS:
        return getattr(self.collection, attr)
    else:
        return getattr(self, attr)