I also faced this problem sometime ago. After some debugging in the L.Popup class, I concluded that the problem happened because of the call to L.DomUtil.setPosition that is done in the method _updatePosition.
This was a really odd bug. After some more testing I understood that problem was related to the fact that L.DomUtil.setPosition uses the css "transform:translate". It seems firefox can't handle well "transform:translate" and the "select" element.
If you really want to use traditional dropdowns in a popup, define "L.Browser.any3d = false;" at the beggining of your script. That will work. But everything else will suffer, so it's probably not a solution. I'm thinking about simulating the dropdown using pure-css. Hopefully that will work in all the browsers. If you do this or if you can find another solution, please let me know.
Here are some more details about the firefox bug:
Bug 664707 - Unable to use drop-down list (<select>) in a div with transform:translate
https://bugzilla.mozilla.org/show_bug.cgi?id=664707