I think there are a few ways this could be handled.
As suggested by Molot, one easy (probably the easiest) way is to have the forest itself exist at cloud level, as forests on mountains might. The forest area would need to have the right climate balance - not too cold (clouds turn to rain) and not too hot (clouds would lift up higher) - and the geography would have to support it with a large, warm body of water nearby... but I'm sure there are many places on Earth that fit that bill, so it's no stretch for it to be possible.
The more interesting challenge is creating an area that does not use clouds, but instead creates its own fog.
One idea along those lines would perhaps be to have a large reservoir of underground water, and trees which have evolved to "sweat" that water out (for some reason). Effectively, this could create a situation where there is nearly perpetual "dew" everywhere to create fog with - at least during the day. The temperature in that situation could, I think, be pretty bearable. With all that moisture there would also either be a lot of rain in the area, or somewhere pretty nearby (as defined by wind speed). The closer the rain is in the area, the more likely it may be for clouds to be in the area to descend on the forest at night to create fog then as well.
Another idea is perhaps an area where trees have grown relatively tall and with quite dense canopy cover. In that area there are also a number of natural springs and areas of hot ground (very hot natural hot springs may be the easier/more likely solution). If the tree cover is sufficiently dense it could do a decent job of trapping the resulting steam, though admittedly it would also be quite hot.