By default forests should always be chopped.
Common rebuttals:
-Lumbermill FTW!: The lumbermill comes rather late in the game. Not only are the hammers worth less, but there's no guarantee that the game will even go that long. They're nice, to be sure, but they're not a reason on their own to save forests.
-Wait till math: I used to think this too, but under scrutiny, it doesn't really hold water. Just like how hammers in the early game are worth more than in the late game, hammers in the very early game are worth more than in the middle game. I would wait at most 8 turns on normal to chop a forest waiting on math, but more than that, and you're seriously stunting your growth.
Legitimate reasons to chop:
-Civ is an exponential game, in that early resources more than pay for themselves given time. An extra early axeman becomes an extra early captured city becomes a number of extra mid-game catapults. Same goes for wonders. The earlier the returns, the better.
-Forests adjacent to the city (or important resources) give enemy units refuge during an invasion or siege. Militarily, it's often best to clear the way.
-Forests stifle resources: river commerce, mine resource discoveries, and even forest regrowths.
Legitimate reasons not to chop:
-Low on worker turns. It's generally best to consider what's best for the empire rather than what's best for the city. Sometimes, your workers have significantly more important things to do, like hooking up resources or connecting a military road network. Generally, you'll want to chop more workers, but this isn't always the case. If a city is low on forests, you won't always have worker-turns available to improve tiles to replace them after chopping them.
-Tile placement. Health is overrated, but sometimes it's important to offset floodplains. Generally city placement should avoid excessive bad health, but leaving the forest to offset excessive bad health is a very legitimate option. Also, plains tiles tend to be worthless for some time unless adjacent to a river, so having those available as 1F2H outs when, again, you're short on worker turns, isn't a bad idea. Forests on tundra as well.
-Emphasis on commerce over hammers. Once again we return to worker-turn priority. If an extra axeman won't help you because you're already paying your army through the nose and your $$$ is so backwards you can't pick up any good economic wonders (or a market or library), sometimes a city will be better off growing on its own while your workers cottage up some other city. A city surrounded by grass forests will flourish on its own just fine without worker help.
-Military defense. If you're facing down a mounted assault, forests can help reclaim your road-movement advantage. I've generally only found this to be true on small fronts.
-No bronzeworking. If for some strange reason, your game doesn't require you to pick up slavery to rush a wonder, get forest chopping to speed development, or find copper in hopes of building axemen or spearmen, then sure, I suppose you could hold on bronzeworking past the time when mass-chopping would pay for itself.
Neat tip: We're all used to telling our workers to chop and then improve the tile, to get the hammers earlier. However, if a city is working a forest you want to chop, it's not a bad idea to tell the worker to just build the improvement. The forest will stay and be worked all the way up until the improvement finishes to replace it.
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