For what I have gathered talking FSS and ATC guys is that Unmonitored simply means that there is no FAA facility making sure that the station is still putting out an accurate signal. When a VOR is unmonitored, the FAA has no real-time idea whether it is working or not but they assume it is still working until a pilot tells them otherwise. Once notified, they will mark it as OTS and send a crew to check it out.

Legally, I have found no FAR, AIM, IPH, IFH, PHAK, or any 135 Ops Spec (company specific) references to any legal significance behind using an unmonitored navaid for navigation under IFR.

Specifically...
-14 CFR 91.181 - IFR Course to be flown says, "On an ATS Route.." or "on the direct course between navigational aids..." with no mention to unmon navaids.
-14 CFR 91.169 - IFR Flight Plan Requirements (alt req's) says mentions the 1-2-3 rule with no mention to unmon navaids either.

As far as I can tell, there is no legal function behind an unmonitored navaid. It seems to be just an advisory for pilots. If anyone has any FAA references that explain unmonitored navaids in more detail, please share!