The prefix of ínstó seems to suggest pressure or movement in a way that minor doesn't, but is that suggestion borne out in their actual use? Quí minátur quasi fíxus est, quí ínstat in aliquem movet?
While unrelated to monere (which is instead related to memini), it seems to me that minari overlaps with it partially in the sense that the threats are warnings. This makes sense since the word comes from minae. It seems to me that the type of threats involved with minari seem to warn away someone (or something) from their present course of action lest there be a harmful consequences.
Thus Lysimachus' threat in Cic. Tusc. 1.43.102 implies that a warning to Theodore of Cyrene:
Cyrenaeum Theodorum, philosophum non ignobilem, nonne miramur? cui cum Lysimachus rex crucem minaretur, "istis, quaeso" inquit "ista horribilia minitare purpuratis tuis."
Surely we marvel at Theodore of Cyrene, a very noble philosopher? When Lysimachus the king threatened him with crucifixion, he said, "Make those horrible threats to those courtiers of yours."
On the other hand, instare also can mean "menace", "harrass", or even "to approach menacingly". This probably comes from its original meaning of "to stand over" or "to loom", with the sense of someone looming menacingly. Cf. Cicero's Cat. 2.5:
ego confido impedendere fatum aliquod, et poenas...aut instare iam plane, aut certe iam appropinquare...
I am confident that some fate looms over them, and that punishments either now plainly instare or at any rate now approaches.
There's no sense of warning here. The threat comes from its immediacy, and that it's in the middle of impendere 'to hang over' and appropinquare 'to approach' indicates that it should be grouped with them accordingly. Their fate is sealed.
Contrast that with the fate of Theodore of Cyrene, who could have turned from his present course to escape crucifixion.
(As a side note, I'm sure that you'll be able to find exceptions. That's expected with playing loose with vocabulary, though.)
Fascinating—especially because, in Italian at least, minacciare ("to threaten"), which I have to assume comes from minárí, definitely has that sense of menace. Feb 27, 2016 at 4:16
1@JoelDerfner Indeed, they're related! Minacciare comes from minax via Vulgar Latin *minaciae, which in turn comes from minae, the very origin of minari.– cmw ♦Feb 27, 2016 at 4:28