Will standing topped off tree trunks work for holding up a building structure?
The main concern here is not the load bearing capacity of the stumps, (which will not be a problem) but obviously the rotting of the wood over time. Essentially rot, is caused by moisture loving Fungi and Bacteria which feed off the wood. So if you can keep the moisture of the wood down, there is no real problem. 

With that in mind, if the stump was enveloped by the structure of the house, and there was no run-off onto that area from rain, I would comfortably say that the stump would last the lifetime of the structure. you can further improve the situation if the wood is of a rot-resistant species like cedar, locust, cypress, oak etc.  and then the diameter of the stumps would also play a part.

If you can, it may be a good idea to dig a few inches around each stump and lay a cement casing around each stump under the ground to ensure that any ground moisture does not affect the wood. 

termites, wood cracking, and new shoots may also be a hindrance if they are not controlled also. 

So the simple answer (and a bit of a cop out) ... If you can control the moisture. it will work. if not. based on the lay of the land, wood species, stump thickness, climate etc... the stump will begin to deteriorate from the word go...