Temples of Gods with a connection to art and beauty would also be good candidates. I would expect there some collected stuff.
-Being a mostly medieval-type setting, there aren't going to be a lot of museums, in the way we have them. I'd expect the few places like that to exist in cities like Waterdeep, Silverymoon, Calimport, however.
But I never heard about real museums, built to exhibit art items, as Le Louvre.
Thanks for quoting this one Green knight !
And if not, you could always 'Ask Ed' over there.
That is, great works of art were/are (in this case) typically commissioned by wealthy patrons, namely the church(es), gentry or nobility. They don't typically share that w/ the public. Perhaps private viewings among "their sort" and so on to showcase their own wealth or power, but not a museum per se.
In other scenaros, specific guilds, religious or secular, might have liberated (read, slain monstrous horde probably) pieces here and there, but again, they would probably be on private display or used in a research capacity.
However, some exceptions I might forsee: elements of Sune's faith being major proponents of all things beautiful might ochestrate a musuem of sorts to share said beauty. It's just "one man's trash..." bla bla bla.
Other faiths (given massive bankrolls) might work in a similar style, but her faith is all about the asethetic so so forth.
Think of museum's in a more abstract manner, not so much a public forum given the chronology and oddity of gaming here. Paid tours of Waterdeep's necropolis might help finance Kelemvore's faith and provide for potters fields and so on. Private fundraisers for arcane guilds and the many wondrous "works of art" (phenomenon, monstrosities, etc) might work in that regard. Drow artisans are always looking to attract wealthy matron mothers looking to flaunt the might of house 'x'.
Limitless possibilities.
For conventional museums in the popular sense, they (if possible) would be in affluent cities, Amnian metropolises, Waterdeep, Arabel, places like that. But they would need extensive security; consider amassing alot of expensive treasures in 1 area w/out the risk of horrible monsters that would kill you so much as armed guards that would rather capture you. It's just financing such security might be an issue. Either way, as crooks or security, endless plothooks for PC.
You could also work in a cannibalized Da Vinci Code sorta' adventure. Big on Knowledge checks, Decypher Script, divination magics, etc. Could be good.
The widespread existence of museums is pretty much a concept of the modernized nations of the world.
A place where already strong security forces are available and where the owner(s) get in contact with other people who should be impressed before they even come to the owner.
So ancient and impressing statues, some portraits, and the like in a great hall in a building of the local government or if someone visits the headquarter of the merchant guild xy...
Another place where you could expect some kind of art, besides temples, would probably be bathing houses and special places sponsored from rich merchants, city patrons and wealthy aristocrats to impress the public (if you know something about the roman and byzantic public baths, they were famous for such stuff).
Anyway, I would expect temples of Sune to showcase pieces of artwork for all to see. She is often a patron of artists, and her dogma is all about increasing the amount of beauty in the world (and sharing it).