A while ago I stumbled across the FAA's exemption database. I 
searched "museum" and planned on posting the 102 results (which cover 1991-2014), but put it off. However, seeing as the FAA will be retiring the website tomorrow, I figured I'd better get on it!
- Cavanaugh Air Museum
 
- Champaign Aviation Museum
 
- Delaware Aviation Museum Foundation
 
- Eagle Flying Museum
 
- Evergreen Air Venture Museum
 
- Firelands Museum of Military History
 
- Indiana Aviation Museum
 
- Heritage of Eagles Air Museum
 
- Kalamazoo Aviation History Museum
 
- Lone Star Flight Museum
 
- National United States Armed Forces Museum
 
- National Warplane Museum
 
- North Bay Air Museum
 
- Northern Indiana Aviation Museum
 
- Olympic Flight Museum
 
- Pacific Coast Air Museum
 
- Palm Springs Air Museum
 
- Planes of Fame Air Museum
 
- Vintage Flying Museum
 
- WarBird Museum of Virginia
 
- Western North Carolina Air Museum
 
To get a direct link to the relevant PDF, enter: 
http://aes.faa.gov/AES/GetPDF?exemptNo=[exemption number]_________________
Tri-State Warbird Museum Collections Manager & Museum Attendant
Warbird Philosophy Webmaster