-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.
Lumbermill has FP=4 (5 with Railroads), which is exactly the same as mine, watermill (w/ PS), workshop (w/ SP) and Windmill (w/ RepParts I think). How are those hammers worse than any other hammers? Or do you mean that they're worth less because they aren't available earlier (which is true)?
I prefer lumbermills to watermills and workshops because they aren't civic dependant - I can run mercantilism or free market and still get the full FP unlike with watermills and workshops. In this regard I would consider lumbermills better than the other two flatland hammer alternatives.
Still, there's only one case where this is the definitive reason to save a forest: tundra forests. Tundra is quite useless in itself, but tundra forest makes for a reasonable lumbermill. Pretty much the only way to use tundra for real.
Now, game over by the time Rep Parts comes? Never for me. I don't play for early conquest / domination, so the game is certain to get to Rep Parts. I'm a builder at heart, not a warmonger. Also, I don't play single landmass maps, which are the only ones where early conquest / domination is truly possible. Nor do I play small maps, not even standard... I play large and epic (would play huge marathon if my computer would have the oomph it takes).
-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.
Admitted, I chop when I need the hammers, only waiting (and prechopping some) if math is close.
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.
The first one I agree with. Chopping early is a big thing, chopping late when your city already has the production isn't a big deal. There's one case where I may consider saving forests for post-math chopping: commerce cities which will have very little production, so I may want to have a few forests to chop some markets or whatever. Chopping an improvement is almost always a better deal than whipping one.
Military reasons don't apply to SP games often.. How often does an AI stack close onto your city really? Duh - they're dealt with when they enter my cultural borders. Rare is the day when I have left a full defensive terrain path though, so it's quite likely that they just HAVE TO enter flatlands before they can get to my city. MP is another beast completely, and one I don't play. I assume things are very different there.
River commerce is one coin, which you get back with the lumbermill. And the commerce is irrelevant in my production cities anyway - commerce cities however get fully chopped for cottage spam (I play CE almost always). Thus not a reason. Forested hills are another matter: hills are the first ones I chop for mining.
And forest regrowth? Almost insignificant. The tile I've chopped is going to get improved soon anyway.
Legitimate reasons not to chop:
Won't go into these more, they're good reasons. I'd add one though: play archipelago tiny islands (I like that map). Your production will simply suck throughout the game, so the one extra hammer the forest provides in itself (add another with lumbermill and yet one more with railroad over the mill) is significant.
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.
This is quite important at times. For me one common case is The Great Library. A city I want that in is unlikely to have high production, and I might not have GE around to rush it. If it has forests (that could be replaced with cottages), I may start "pre-cottaging" the forests, finishing the job when I count the chops to be enough to complete it, maybe getting as far as dropping half a dozen cottages out of the forests in just 2-3 turns (depends on if I had the worker turns to precot that many forests).