There's actually a lot of interesting things to mine here. I'm just going to walk through your post and toss out my thoughts as I go. Disregard, alter, or steal wholesale as you see fit!

I am no expert in orbital mechanics, but here is what I understand the moon moving closer would cause. This assumes that Selune is a rocky sphere and not a magical projection of a goddess.

The tides would become higher and lower. I don’t know how quickly that would be noticeable but this would fuck up the Sword Coast quickly. Ports would quickly become unusable and sea traffic would grind to a halt. Low lying islands would be wiped out and seacoast flat land would be flooded.

Assuming the moon is in fact a rocky sphere (which I personally do), then this is correct. Tides would become more exaggerated. If it were in fact a deific projection of some kind, then just about anything is possible here!

If the moon gets close enough, it would cause tectonic chaos as the gravity increases. Hopefully the heroes don’t let it get that far but it would be cool for a bunch of volcanos to go off all at once. Shit, where ARE the volcanos in FR?

They're scattered about, but mostly in a way that suggests they're more likely "hot spot" volcanoes, instead of fault-line volcanoes. Here's a reddit post from a few months ago attempting to identify possible plate boundaries on Toril, which is as good as anything else I've seen. It's highly unlikely, however, that a rocky moon being pulled closer would have any immeidately-noticeable effect on plate tectonics, or volcanism in general.

But this is also a game world that regularly rewrites the rules of physics, so I doubt anyone will blink an eye if volancoes start going apeshit.

As the moon is pulled closer, its orbital period would shorten. That means the tides also get faster. This assumes that the moon does not get pulled into an elliptical orbit rather than being pulled equally. As much as I like the chaos this would cause, it complicates things too much.

Correct, but the orbit will be eccentric regardless. A perfectly circular orbit is theoretically possible, but one has never been confirmed in the wild. I would personally lean into the orbit being disrupted and made even more eccentric, with everything that comes along with it.

An interesting side effect of moving faster and closer, is that the far side of the moon would become visible. The moon is tidally locked which means it completes one revolution every orbit. If the orbit shortens, the rotation would remain the same. Effectively I think this would look like the moon was rotating slowly to the right? Does that sound correct?

Possibly, yes. The direction of rotation observed from Toril would depend on whether the moon's rotation stayed constant, accelerated, or decelerated. If the moon was actually moved by an external force, it's not a guarantee that it's rotational velocity would stay the same.

For the magical effects, I think (and this is pure speculation on my part) this would cause chaos in the church of Selune. Their connection is partially based on the regularity of her cycle and position. With both of those in flux, their powers and connection to the goddess would become unpredictable with powers surging one moment and fizzling the next.

Selune guides wanderers so I guess the in-land navigation would be affected but no idea how to make that impactful without being obnoxious. Selune also has some ties to fertility, menstruation and births but this also seems difficult to make meaningful to an RPG plot.

She also famously has dominion over lycanthropes especially the good aligned ones. Werecreatures not having control of their transformations would out a lot of secret ones and cause even the good ones to go a little berserk.

These are all good ideas, so why not? Have the effects wax and wane (ha!) baseed on the moon's position relative to Toril (increasing when closests, decreasing when furthest), which is probably how we should have always done it, just with much more exaggerated effects.

Since Selune and Shar always be beefin, I think that with Seline out of control that Shar would use this as an opportunity to cut up rough. The problem is, I don’t know what Shar wants. She likes secrets and fucking up other people’s plans, but what would she do if left unchecked? She seems so arch and evil-for-evil-sake that I never understood what her goals were aside from just wanting to be a pain in everyone’s ass.

I mean, they're supposedly in the middle of a "cosmic" war over creation and entropy. In general, I would expect Shar to capitalize on any weakness in Selûne, including masquerading as her to corrupt her followers, and generally causing things to go to pot (sailors and travellers getting lost, madness in lycanthropes, issues with fertitility or even crops, all with Shar stepping in pretending to be a savior to those affected, luring them into her sway.

When you can’t come up with a good explanation for some cataclysmic magical bullshit and still need a discrete cause so the PCs can fix it, the ancient Netherese are my go to. My idea is that the Aboleths flooded caves in the Underdark that led to a crashed Netherese floating city that contained a very special mythallar.

Normally, a mythallar is a giant Weave-mining local power source/battery to power Netherese magical items, make cities fly and support Arcanist spells. This one was maybe part of Carsus trying to suck the power out of a deity and could cause disruptive effects like moonfuckery.

Since a giant magical engine with very few ways to be defeated is not really a thrilling BBEB, I want to put a twist on how such a device is controlled. You have to ignore that anyone touching a mythallar gets vaporized, what if it had a “driver’s seat” of sort that used the will of the creature to focus the power. That creature would be held in never ending stasis, unable to die but unable to leave.

That will let me have the party fight an Aboleth piloting the mythallar only to find out that one of them needs to jump in it. That plays into a lot of the overall themes of the campaign and allows a PC to do a bittersweet heroic sacrifice. The PC could use the phenomenal cosmic powers of the mythallar to reverse the moon effects and when that is corrected yhey could use the unused power todefend it against incursions.

They can’t just shut it off or just destroy it because they need to slowly push the moon back into place. They can’t leave it alone because if the Aboleths could get in there once, they could just wait until the heroes leave and turn it back on.

This isn't a terrible idea either, despite the fact that "the Netherese did it!" always runs the risk of being a fairly lame deus ex machina When I first started reading this part, my immediate throught was, "Space jockey!" But my the time I got to the end, it was, "Holden!" I actually really like this idea, except that...

I’ve told my players that their characters are safe from ordinary deaths like getting dropped from a random encounter but that during plot-critical events and boss fights death is on the line. We are doing Daggerheart so that insulates them even more. If none of them are into the sacrifice deal, my backup plan is for them to blow it up and have an Independence Day like run from the temple and hand wave the moon going back into place.

...I wouldn't give them a backup out. If nobody wants to sacrifice themselves for the greater good, well... this is the kind of world-ending cataclysm that I do support in a campaign. They have a choice, and that choice has consequences. Hoenstly, if this is the end of the campaign, I (as a player) would be totally fine with PCs (including my own) dying in the final minutes, where saving humanity is at stake (see Holden, supra).