How much weight will a 4"x8"x20ft oak beam support?

The strength of the wood beam will depend on the specific species of oak, the grade of lumbar, whether there are any cantilevers, side bracing, and other factors. But 20 ft seems like a very long span for a 4x8. Glancing at some load tables I would say you're in the ballpark of only a few hundred pounds capacity, which is not sufficient when you take into account the weight of the structure, the weight of occupants, snow, etc. (Don't take my word for it: you must get a more accurate number before you start building). (By the way, where are you going to get a piece of solid oak that long??)

Another side note: you don't say anything about the design of the tree house but be careful about how you connect the trees. When the wind blows the trees will move independently and you need to make sure that your structure isn't rigidly fastened to both trees. There are many examples online about how to make a platform that is fixed to one tree and slides at another. If you don't allow for movement of the trees you risk damaging the treehouse or the trees themselves when a gust of wind blows.

I would suggest you contact a structural engineer or local building code office. They will be able to provide you with load bearing numbers for common lumber, i.e. spruce-pine-fir; your oak timber will carry considerable more weight. When we were kids (back in the sixties), the material used was limited only by what we could scrounge from scrap lumber piles, filtch from neighbors, or drag home from a construction site. Our treehouses were no architectural marvels, but nobody ever died from the treehouse collapsing on us either. But that was a day when three nine year old kids had more sense that a lot of adults today. Just have fun with it, and let your kids help.