Ok I am totally confused, does cutting down trees aid or hurt your civ? I thought it helped production because everytime it tells me Im getting like +30 hammers or something...
Short term gain for potential long term losses.
If you have enough hills to give you production, you probably won't miss them. If you're in a grassy/watery area, you'll probably want to keep the trees around.
In Civ 4 having a city near a forest also provide a health bonus to the city which was quite important, and it also provided happiness with certain civics. Civ 4 also had the short term production bonus.
In this game cut the trees down if you feel like it for for some short term production, or save it up for when you're building a wonder in that city. The poster above is right as long as you have hills for mines there is no long term penalty.
Sigh, another example of how dumbed down this game is compared to Civ 4...
I usually play as the iroquois so i advice against it.