Our museum moved to free admission about 8 years ago now, and it worked out wonderfully.
In the spirit of transparency, though I should note that we are a local, small-town museum. The admission price was ridiculously cheap (like 50 cents for an adult), and the museum was not regularly advertised as being open...so free admission alone may not be responsible for all the changes.
However, since implementing (and advertising regular hours) attendance has seen a dramatic upswing and so has our income--people are much more likely to drop a $10 or $20 into the donation box. Even if 50% of guests don't donate or don't donate as much as you would have previously received from admission (depending on what that was) I'd say about 50% will give more and balance that out. We've even had a handful of visitors drop $50s and $100s in, saying they appreciated that it was free for everyone. You're also more likely to attract families and large groups, as even $5 or less per person may be cost prohibitive to them.
Advertising that it's free admission is a huge boost for sure, but it's also a great thing to mention on grant requests too--with free admission you truly can represent the whole community, and grant boards will recognize that.
So, in short: Attendance skyrocketed, revenue skyrocketed (but like I said, 50 cents per person isn't gonna get you very far), and membership has increased but I should note that we don't necessarily have "museum" memberships...instead our parent historical society offers them, and since the museum itself is free there isn't a direct benefit of free tours to members so I'm not sure if there's correlation there.
American Alliance of Museums2451 Crystal Drive, Suite 1005Arlington, VA 22202