I've just scanned quickly over the posts, so forgive me if anyone has mentioned "heave" before. If the tree is removed, all the water which it used to soak up from the ground, and that'll be a lot of water, will be left in the ground, which CAN, I'm not saying WILL, swell up and actually have the reverse effect of subsidence, ie the ground CAN rise.
Seek professional advice , perhaps the Tree preservation officer at your local council. Tree surgeons are more inclined to say , have it out, but the TPO gains nothing from giving impartial advice. Once it's gone, it's too late to change your mind.
As my dear late mother would have said " Act in haste: repent at leisure."