How long could a person live in a treehouse?
Theoretically, a very long time... depending on the weather.
I think the primary issue with living in a treehouse would be heat and insulation. Its not like a regular home. Its more like a tent in terms of its insulation capabilities.
However adding extra insulation to a tree house is not impossible. All it would take is some carpentry and construction skills, and knowledge of how to insulate.
In my opinion the easiest way would be to make seamless walls out of wood, an outer section of wood (with all holes caulked), and then insulate between the two sections with an insulating foam which hardens and forms air bubbles.
Heating wise I'd recommend lanterns that run on kerosene, candles, etc, both as a source of light and warmth. I recommend using bricks or stones to create a stone area on which to safely store lanterns / candles.
PlumbingIn theory you could collect rainwater and use that for cleaning, but realistically there isn't much point in doing that inside a treehouse. It would make more sense to wash everything in a nearby river, one with exceptionally clean crisp water.
For expelling bodily fluids building an outhouse on the ground level seems the logical route. I recommend designing it in a way so that rainwater can be collected for flushing/controlling the smell and storing the septic material deeper underground, preferably in some kind of metal tank.
FoodSome people might prefer to use such waste as fertilizer for growing plants. Ideally I think this should done by building a greenhouse, but a person could also make a traditional outdoor garden and make the effort of weeding / making sure insects/pests don't eat all the produce.
Water, DrinkingIt should be noted, due to bacteria, that you should not be drinking rainwater or stream water. Thus what I would recommend instead is natural juices (apples, orange juice, grapes, etc). At least then you know it will be safe. Otherwise what you end up doing is having to buy clean water...
Or get a water filtration system, which means you need electricity, and a source of water (a well preferably).
Check out the following video:
http://www.youtube.com/watch?v=98h086DWLaY