A common problem caused by tree roots is blocking drains. They do not usually cause the initial damage to the drain and will only enter drains which are already damaged and leaking. Therefore, if drains are watertight, roots should not normally affect them. If your drains are blocked by roots you will need a drainage company to assist. It is possible that the drains will require lining or replacing. Removing the tree seldom resolves the problem as the drain remains damaged and can leak (possibly causing foundation damage) or may be infiltrated by the roots of other plants unless the drain is repaired.
If your driveway or drains have been affected by roots, this does not necessarily mean that your house will be damaged next. Root damage to buildings is much rarer and is usually caused by subsidence: an entirely different mechanism to driveways and drains damage.