If Firefox is maximized then make the window smaller and drag that window over to the correct monitor and close Firefox to save that setting.
After restarting you can maximize again.

Un-maximizing, then dragging the window, then closing Firefox doesn't fix the problem. Next time I open Firefox, it's in the same old position.

This can be a problem with the file localstore.rdf in the Profile Folder.

Delete localstore.rdf or rename the file to localstore.rdf.sav in the Profile Folder to test if the file is corrupted.
See http://kb.mozillazine.org/Corrupt_localstore.rdf
(caution: do not delete the localstore.rdf file in the Firefox program installation folder)

Note:
Deleting the file localstore.rdf will reset the customizations of the toolbars to the defaults.
You can rename "localstore.rdf" to "localstore.rdf.sav" to test if that solves it.
Then you can restore the customization by copying "localstore.rdf.sav" to "localstore.rdf" if it didn't work.

Win7, mostly in XP mode as I don't like aero.

I move Firefox to the monitor I want.

Expand it to full size. Grab title-bar and bump it against the top of the screen. When the outline appears, let go.