The Henry Ford museum in Dearborn has a small fleet of Model Ts that they use to give rides to visitors. The Gilmore museum in Hickory Corners, Michigan does the same and even goes another step further...they'll let you drive and even teach you how if you've never driven a T. They also routinely exercise many of the vehicles on display.
The Crawford Auto and Aviation museum here in Cleveland also takes their vehicles out on the road. They have several at the Cleveland Auto show this week. Several years ago a friend was one of the docents and I helped him and others move some of the cars from one location to another. It was great fun!