For new musicals:
The shortest-running Best Musical is Passion at 280 performances. It's post-Tony run was 30 weeks. Just ahead is last year's winner A Strange Loop with 293 performances. 31 weeks of which were after the Tonys. Hallelujah, Baby! also played 293 performances but, as ablankpage noted, those were all pre-Tonys.
I haven't gone through all the winners but there are undoubtedly several from the 50s, 60s and 70s when shows would regularly open almost a full year before the Tonys. One example would be Bye Bye Birdie which opened in April 1960, won the Tony in April 1961 and then closed in October of that year. Fewer post-Tony performances than Passion or Strange Loop but more overall.
For musical revivals (not including limited runs):
2005 - La Cage aux Folles closed 3 weeks after the Tonys
2007 - Company closed 3 weeks after