I'd really love to start writing Honor Among Thieves fic, but all of my experience with DnD is from playing homebrew. Do you have a recommendation for a good place to start learning about the lore of Faerûn?
*whispers* I have a confession to make.
All of my experience is playing homebrew.
Everything I learned about Faerûn I learned from the Forgotten Realms wiki. I just started by reading pages on Szass Tam and the Red Wizards and so on, and followed the rabbit hole from there. I studied a downloadable map, too, helpful for place locations!
As a result I highly doubt my knowledge is as good as, say, someone who's read the actual D&D novels set in that world. But it's been enough to get me by!
Luckily, I don't think anyone is expecting you to be an expert. Those of us who do play D&D who are in the fandom by and large seem to be understanding that there are a lot of total newbies who don't know anything beyond loving the movie, and I don't think anyone will be upset if you don't throw in a ton of lore. In fact I've had a ton of comments on Untranslatable Shadows complimenting me on my dungeon design, unaware even with my fic tags that I'm modifying the infamous Tomb of Horrors dungeon.
Feel free to just start looking stuff up on the wiki and see what you get. Do a little exploring, go down a rabbit hole or two. You can also come up with a story and then do some research to see what will compliment it. You can even just keep the setting of your story in Neverwinter and the local surrounding area.
So that's what I'd recommend - dive into the wiki and see what you find.