Does anyone think there are particular authors or classic works that lend themselves to being read aloud? I already do plenty of reading, but I want to replace the mindless music listening I do at the gym with some audiobooks. Any suggestions?

Most poetry of course! Oh, and Middlemarch, Moby Dick, and The Remains of the Day spring to mind.
I've loved reading Ulysses and anything by Virginia Woolf out loud
If you like him, I really enjoyed the DFW audiobooks for Consider the Lobster. Not sure if he's done more as well
Maybe it’s not quite what you’re looking for but the A Song of Ice and Fire audiobooks are pretty good. Roy Dotrice does the accents well and the books aren’t super high brow that you need to read and concentrate to understand properly.
definitely anything by Nabokov