Oh, Very Very Yes
If you want some dead gods, come on down to the Avatar Trilogy, where Bane got Banedead, Bhaal rolled over, Myrkul went from being God of the Dead to dead of the god, and other such fun events. They're not the only ones - just some of the many who had themselves a very bad day during the Time of Troubles. Each one had the bad luck to be forced down to Toril in mortal form (just like all the other gods) by Ao and catching a slight case of death. Mystra (not the current Mystra, her predecessor) was another of the more notable deities who went kaboom.
This was all part of the changeover from 1E to 2E in Realmslore, the rationale for why the world (read: the rules) ended up slightly different. Even before them, however, gods have died on Toril. Sadly, most of them did so long ago and offscreen - Moander the Darkbringer, Tyche, Murdane, Auppenser, Aumanator...
What happens can vary - in the case of the Avatar Trilogy, the responsibilities of the dead gods were taken by new ones. Sometimes, like Ibrandul, this meant an extant god eating the portfolio of another. Shar did that. Mystra and the Dead Three above had their duties given to ascended mortals - Midnight the mage, Cyric the rogue, and later Kelemvor the warrior. The tasks have to fall to someone.
At the local level
...you will get the divine essence of the dead god contaminating the area. Hopefully they're not substantial and this can be contained in some way - nothing says "salting the earth" like exploding a god of the dead and letting his vapors get all over the place. If you have Auril getting murdered, and she's properly dead as opposed to being able to recover herself in the Outer Planes, you can anticipate some kind of horrible wintry outcome for wherever she falls - quite possibly in a rather permanent sense, which could have interesting effects for your world. A miles-wide patch of eternal winter would change not just the local climate, but also have lasting repercussions on weather patterns, migratory patterns, water tables... given a decade or so there could be some widespread alteration to an entire region of the Realms.
On the jobs front
...someone needs the portfolio of Winter and Cold. Will it be Umberlee, the cruel ocean goddess, taking on a frosty mantle and spreading deadly icebergs throughout her seas? Might Malar step up to the plate, commanding the harshest of seasons as well as the most rapacious of beasts? Most likely would be Talos, her direct superior and the god of storms and destruction.
Of course, it could also go to someone else entirely - a mortal, perhaps, who has proven their worth. Said worth, it must be noted, does not need to be "good" worth - Ao is totally neutral and would be willing to accept a candidate who is a cold-hearted son of a goat. Eliminating the Frostmaiden may well result in someone much worse taking her place.
There's also the oddball possibility - the sleeping Ulutiu, god of the Great Glacier, might stand to inherit some of her job. This would almost certainly wake him, causing the Glacier to begin to change as he rouses from his torpor and riling up the gods of the giants, who remember that Ulutiu has a poor history with their dad and mom.
In general, a god does not die on the Material Plane. Their aspect might, or their avatar might, but a god is almost never personally on the Material Plane and there are dire consequences to being so. Even if slain, their essence should retreat back to where they have most vested it - their home realm on the Outer Plane. For Auril, this is Winter's Hall in the dreaded and windblasted tunnels of Pandemonium. Fun place. Definitely visit.
Assuming she's not 100% absolutely dead when faced on the Prime, her would-be murderers are probably going to want to hunt her down and finish the job while she is relatively weaker than she will be once she's recharged. That will put them in her home turf, where she will be at her most mighty and secure, surrounded by servants and in her full and terrible glory.
Once she's finally toast, however... well, besides finding out who takes her gig, there's also the fact that she shares her place with a neighbor - Loki, the Norse god of mischief. Come to think of it, he might even offer help in the endeavor - though anyone with a lick of sense will think twice before blindly accepting.
It should be noted that while the portfolio needs to be held by someone, it doesn't necessarily need to be exercised in an evil manner - Kelemvor, who inherited the job of death god from some real jerks, is steadfastly neutral in demeanour. Finder Wyvernspur inherited the godhead from Moander the Darkbringer, one of the most evil deities Faerûn has ever had the displeasure of knowing, and turned his portfolio of rot toward the principles of renewal - something new growing out of the old.
As for her realm - Auril is lucky that she often has Loki around, but he's technically allowed to go to Ysgard as well. If there is no godly essence in residence, Winter's Hall will begin to decay in a way that resembles an implosion, a rather horrifying spectacle that takes decades to resolve but leaves a nightmarish stain where it sat in the cosmos. For bonus points, Auril's realm happens to be linked to the headwaters of the River Styx - anything that disturbs the flow of the great lower planar river could have severe repercussions on the Blood War.
That is not dead which may eternal lie...
Now, of course, the Realms have also seen their share of gods coming back from the dead. Leira, Goddess of Deception, is - well, who even knows if she's alive, dead, or a duck in a parka? Bhaal's grand plan for his own resurrection spanned two really good video games and dozens of offspring. Bane used just one - Iyachtu Xvim, who got bodyjacked by his dad. The best plan for recovery, however, is pretty darn simple:
As much as gods don't appreciate being stabbed, fireballed, poisoned, or whatever else it is you plan to do to them to make them stop being quite so personable and ambulatory, the only true way to expunge a deity for good and always is to end worship of them. Their corpses float about in the Astral Plane until they receive sufficient veneration to renew their conscious existence. You can try to do this by eliminating the worshippers themselves (inefficient) or by converting them (generally more successful).
The best module for godslaying...
...might still be the original, though: the campaign against the Demon Queen of Spiders herself, Lolth, in Queen of the Demonweb Pits. This module depicts how tough it is to actually chase down and end a deity on their own turf. It's aged and at times quite bizarre, but it's still the most classic culmination to a godslaying campaign, and the drow goddess is a Faerûnian favorite.