Based on the show itself: Once American Idiot Motown Marry Me a Little
Based on the cast album: Spring Awakening The Bridges of Madison County Most things by LaChiusa (though I don't know if they are so much boring as they are excruciating)
I have walked out of two musicals out of sheer boredom.
One was The Civil War, the other was Shogun. Hon mention: Chu Chem (which I somehow sat through). Then there's Memphis which I didn't see on stage but couldn't get beyond 15 minutes on Netflix.
In terms of the most boring musicals that weren't complete flops of that ilk (or Memphis), as much as I hate to agree with After Eight about anything, Passion is at the top of my list.
I will be honest here and say I fell asleep during the most recent revival of How to Suceed with Darin Criss, it drives me nuts when I see people do this. I had a very long day, it was January and freezing and got TKTS tickets around dinner time. Even with discounts those were the cheapest seats they had, they were in the far orchestra off to the side . I've never had any interest in it and went as a favor for a friend who was coming into town because she wanted to see him, never mind the show. The worst $100 I ever wasted. Dirty Rotten Scoundrals isn't one of my favorites either. The best part was my friend agreed with me and she thought it was boring.