How to kill a God?

I’m running a campaign where my players are close to discovering an organization who believes the Gods don’t do enough and aren’t worthy of their power, so they are killing off gods. 

My question for y’all is how would you kill a god? What would need to be done or take place to kill a god? What might the repercussions look like on the world?

These answers partly depend on how gods work in your world, but I assume you've taken the sensible approach of not worrying about it until it became relevant.

The first things that come to mind for whacking gods are "they have some kind of god-killing artifact" and "they have control/connection to some kind of entity that can kill gods". (For the second one, it could well be another god in a funny hat, or it could be something outside the realm of the divine.) A lot of D&D worlds have gods gaining power from their worshippers, and the way to kill them is to wipe out their followers, but that doesn't really work for the narrative you're going for.

As for the repercussions, the most obvious is that the god's worshippers will lose access to their powers. (Note: don't do this to a player without some kind of quickly-available power backstop --  losing access to your character abilities for any length of time is not fun.) 

It's also going to be a source of political upheaval. Religious organizations are going to be movers and shakers in most worlds, and when they lose backing, other powers, both secular and religious, may look to take advantage. They won't immediately lose power, but it'll be a big shakeup, particularly where they're enmeshed in, or are the power structure.

Further consequences are up to you. Do the gods actually make the world run? If you kill Apollo, does he no longer carry the sun across the sky in his chariot? Or does it continue onward? Or are there slower, but still potentially catastrophic consequences, like the fires of the sun starting to slowly fade? 

Also, if they do kill a god, and there are world-altering consequences, there's going to need to be a way to fix it. (You don't have to, and probably shouldn't, have a solution in mind right now, but you need to put some elements that the PCs could use into play.)

My go to is a knife made of cold iron, quenched in the blood of an immortal, and tempered with anti-magic.

Really it depends a lot on the kind of power scale you're writing from. For typical D&D lore, you're looking at something in the tier of Olympians, so realistically mortals have no shot- they're generally unreachable, can readily find and smite mortals who are getting too uppity, and it usually takes special plot devices just to inconvenience them. Obviously this isn't the only model, but if you're doing anything along the classic D&D lines, it begs the question of how they're keeping the gods from becoming aware of them and taking direct retaliation.

The most common handwave in D&D against divine interference is that the pantheon(s) collectively have some code of conduct that limits actions so they don't end up all constantly stomping the sandbox flat, but a common credible enemy would logically be something that'd create enough consensus to get past that- though there is the workaround that the pantheon is in fact so badly dysfunctional/complacent they don't take the threat seriously until they're on the back foot.

Honestly, I tend to find the "gods need prayer badly" trope doesn't really track to D&D- the usual premise is the gods came first, arranged the setting, and then populated it, so why are they now terminally dependent on followers when it took a lot of divine heavy lifting to make them in the first place?

And even minor deities tend to have tens of thousands of followers.

Depends where they get their power from, but if I were the head of an anti god faction, I'd snuff out the worship of said god. Presumably their power comes from the worship of them. There is actually a real world example of this, not the active suppression of deities that were believed to exist, but in Soviet Russia they persecuted anything remotely religious. In it's place there was a cult of personality that formed around Lenin and those who came after him. Which I imagine would be a pretty effective way of shutting down a gods powers while empowering the head of state. At that point, in fantasy land, enough mortals might be able to kill said god while they are weakened. You could probably even tweak it so the head of state is almost worshiped themselves ironically. 

Eh, like I said, given that in most settings the pantheon are the ones who created the world and populated it with mortals, it's rather counterintuitive that they can't exist without active worshippers. Plus they typically are considered to be connected to natural forces or concepts like love, murder, etc. and it's a common point that they draw strength from that as well. The premise outlined above is basically what happened with the God-Emperor of Man in the Warhammer 40K-verse, except it was doomed to fail against the Chaos Gods because they're fueled by their concepts/spheres rather than active worship, and thus basically can't be killed so long as there's intelligent life in the 'verse.

Now, trying to arrange an apotheosis so you've got someone who can go toe-to-toe with them is more viable, but that doesn't really fit the initial premise of a group that's systematically killing deities, and as I noted the thing with typical fantasy deities who follow models like the Olympians or Aesir and Vanir is that they generally have the power and awareness to bring the hammer down on mortals trying to get uppity, and the usual justifications for why they aren't more involved in a campaign don't hold up once the group becomes a credible threat unless they've outright got an instant win god-slaying superweapon, which then raises the question of how a party of PCs has any shot against them.

It's not impossible, but it takes either a group that's good to roll with the adventure as presented rather than pick at the details of how this ball got rolling and why they're in a position to deal with it, or what's probably an unreasonable amount of time pinning down the mechanics to have a plot that stands up to scrutiny.

I'm presuming this person is asking from the perspective of the Forgotten Realms.

As I recall gods are not allowed to interfere with mortal affairs due to alleged planetary destruction and it'd also piss of Ao who is the god of the gods. I always thought it was a dumb explanation but there you go, it's kind of in the lore that gods can't actually directly intervene. 

So if you're going Forgotten Realms, the gods can only push back in indirect ways. That said I don't know if this is an offshoot of the Forgotten Realms or something entirely 100% homebrew. Never really got specifics. 

You're missing the point- it's primarily an understanding, not an absolute cosmic law. We know for a fact that gods do intervene directly in exceptional circumstances of one degree or another- look at BG3, Llolth's whole deal of micromanaging the drow, Tyranny of Dragons involving an attempt to summon Tiamat out of Avernus, etc. Deicide by mortals is about as exceptional as it gets, so even in FR lore it'd be within the body of established material that this would draw a direct response, particularly if there's no god or similar tier being on the other side who can escalate things into a planet-cracking fight.

It depends on who's adapting the Forgotten Realms lore. That's just the truth, you got plenty of people who nitpick the lore in that game. Like, I don't know would the gods intervene? You can absolutely make the argument for that, just as easily as you could make the argument they couldn't cause AO said no because hey maybe it messed up the cosmic balance. So hey, the mortals are handling it instead because it's safer. You know it absolutely can depend on who is adapting the lore. It really isn't this black or white thing you're making it out to be.

Also I don't remember ANY god in Baldur's Gate 3 directly intervening at any point at any time during the whole campaign. The closest I saw was Mystra sending her messenger boy to lecture Gale about how he needs to be a living time bomb. Like am I missing something? It was absolutely an all hands on deck situation during the whole campaign. 

I mean, if we're playing the "it depends" card, then literally anything is on the table, so that's not really a constructive point. Within the FR, there are multiple examples of gods bringing about some form of overt divine intervention, so as I said you need a group who's ready to just take "because the plot says so" as an explanation if we're using the FR or an analogous setup and the gods are just sitting around nudging followers along while some cult goes around killing them, particularly if the cult lacks a ringer of similar caliber to cover them on that front.

That's exactly like Baldur's Gate 3 does though? Like, the example you bring up yourself. The gods literally tell all the mortals to take care of it themselves.  Which begs to question, when are the gods actually allowed to intervene in a way that is more than just, "Okay I choose this mortal as my champion." 
When Karsus stole Mystra's powers and tried to become a god, her response was to sacrifice herself by removing all access to magic. Like, even in these doomsday scenarios like when a god has her powers stolen, their best response they are allowed to muster is to just sacrifice themselves. 

I'm looking at like, the most exceptional examples in DND lore and the most extreme action they ever seem to muster in regards to mortal affairs is to sacrifice themselves to continue to perform their primary assigned job (overseeing magic and the weave). The gods don't touch mortal affairs, even in the most extreme circumstances their responses are lacking. It seems to me the only time that gods can intervene is when their assigned role is at risk. (Karsus literally caused irreparable damage to weave which is why she acted, it wasn't even about Mystra herself losing her powers)
Past precedent shows they are extremely limited in how they can actually interact with mortals. The lore about AO was likely written to justify why mortals literally take care of all the world's issues.