Some add-ons interfere with the Add-on Manager itself, causing it to open very slowly or not at all.
Can you please try opening Firefox in safe mode and then trying to open the Add-on Manager again? My bet is that the Add-on Manager will open quickly when you do this.
If the Add-on Manager does open quickly in safe mode, you might want to return to normal mode and then disable add-ons one-by-one until you find the culprit. If you have AutoPager or Tab Mix Plus, try those first, as they have been known to cause this problem in the past.