Interesting. I'm a full time Firefox user as well and honestly can't think of an example where this happened to me. There have been a couple times where I was warned my browser wasn't supported, which is stupid, but it usually works fine anyway.
I've definitely had sites where FF doesn't work properly with LastPass. I've already had sites that plainly don't work at all with FF. I've never blamed the browser for that though. It's the developer responsibility to make sure I can access their site or I go elsewhere.
To me, if the site doesn't work on Firefox, the site is broken. End of story.