Come on, this is an easy question. Ask yourself what kind of city you want to have, and that will give you the answer to your forest question. If you want lots of production, leave them up for the most part (especially in the early game), if you'd rather have pop, or rush build improvements, chop them down. They're kind of like hill/mines for cities that aren't on hilly ground. The only exceptions to this are forests on hills and forests on grassland (at least for my play style).
When they're on hills, in the early game, a mine will provide you with one extra shield and give you that production rush for chopping it down. So to me that seems like an easy choice. When they're on grasslands, I tend to leave them , because that's a tile with 2 food, one shield, and the problem I normally run into is that I have too much population and not enough productivity (still trying to break my old civ habits) - so I'll leave that forest there rather than have a tenth tile that gives me 3 food, 0 shields. Plus, as I progress in the game and get lumbermills and railraods, that tile that might have given me no hammers, now provides enough food to sustain the city pop using it, and 3 shields as a bonus.