I see no logical reason to use FoL if your not elves, seriously ancient forests<ANY other improvement, the ancient forests it wont help tile yields because there are other improvements that destroy the forest that would give better yields, sure it gives a little extra defence but I still see no point. ESPECIALLY because you cant even build lumber mills in them. Besides their priests and heros are not that great compared to other religions.
Hiphopin, have you looked up the Guardian of Nature Civic? Give it a looksie.
Now, let's consider that you have a city with a number of forests in its radius of say, 6. That's a fair bit, but not overwhelmingly large.
That leaves 14 squares you can work that aren't forests. You'd need 14 total happiness (+ more for war weariness, liberty penalties, or whatnot) to work those 14 squares, and enough food to as well.
Like you say, these squares can probably outproduce even an Ancient Forest (Though farms don't really, but cottages and mines do). So they are probably the ones we really want to be working.
Now lets factor in Guardian of Nature. This civic will give our test city 6 more happiness, and 5 more health. At the point when you can first get GoN, you may have a happiness cap of around 7 or 8 (As in, the number of population points per city that can actually work tiles is 7 or 8). Maybe higher if you get some lucky early game resources. Of those 14 tiles you were working only 8. With GoN, you get a big boost to 14, and have room to grow!
Here's where it gets cool. Not only will you have room to grow, but because your forests start producing 3 food as ancient forests, you can work these tiles for a while, cause extremely rapid growth (At size 8k, if you work 3 Ancient forested grasslands, and 2 grassland cottages, you'll be making +6 food per turn). You will also have a good solid +5 health from the civic to maintain that rate of growth.
The overall boon of FoL, is the combination of all these effects. Consider the output of your empire. All your cities rapidly grow from size 8 to size 14. Some in deep forests will grow even more and faster. You've really just increased your output by about 80%. That's production, commerce, everything.
80% out is huge, and extremely worth it. When you work the numbers for FoL, it is often the strongest religion in the game, and it is a shame more people don't realize this.
Oh, and remember, if you want to, you can still chop down ancient forests and put up other things.