I started playing a game with the "automated workers remove features" set so that my forests/jungles won't be removed. Aside from missing out on the production bonus of chopping them down, is there any disadvantage to leaving forest or jungle tiles?
It's mostly about what you want your city to get. Is your city already surrounded by hills, giving it ample production? A farm is a good candidate to replace the forest, especially if it's over grasslands. If you need production or want gold, however, leaving the forest in place is a better idea, for either a lumber mill or a trading post.
Regarding forests on hills - it's probably better to replace them with mines. You get a little less food but it's better when you try to focus your city on production. Also the production boost from cutting down a forest is useful in the early game.
Jungles are generally a "bad" tile - like tundra - but in my personal opinion, up until the late game it's worth leaving them for the science boost they give once you get a university. Just make sure you leave enough production and food tiles available to the city.
Once your science per turn is in the hundreds, though, you might consider removing them.