It's certainly well known that trees consume a vast amount of water. When this "abstraction" of water ceases, it will of course mean there will be more water in the area. The consequences will depend greatly on the type of ground you have. Naturally, if it's well-drained, then most likely there will be little effect.
"Clay heave" is possibly the most common problem. If saturated clay freezes, then it can push upwards on foundations. Even without freezing, swollen clay can produce something similar. This is a possibility.
Weighing up the pros and cons, I think I would prefer to not have the possibility of interference with the footings by something as powerful as a large tree root. I doubt if you want to do this, but a small excavation trench at the edge of the building would tell you how far, if any, that the root has got into the foundations.
As Sqad would say.......... monitor the situation. I've seen plenty of large trees removed with no adverse effects.
Any idea what your local ground type is?