Let’s start from the worse case and work back to what you should do
While laws vary from place to place in most cases if a tree falls down it is considered an act of god and the it becomes the responsibility of whatever landowner the tree falls on basically if it falls and hits your house (your insurance and not his covers it and your insurance premiums got up not his) if it falls in your yard you have to clean it up not him. There are things you can do to change this.
OK now we get to what should you do send a certified letter stating that you fell the tree is dead and a hazard, this is considered putting him on notice and in most places this changes the legal liability so now he knows there is a problem and if something should happen now a court might find him negligent or failure to take action now the tree falling becomes his problem. If you want to strength your case first take a ton of “good” photos of the tree and for a slam dunk get a tree profession to state in writing that the tree is dead and or is a hazard. That is a legal CYA
OK now how to go about taking it down, send the certified letter wait 15 days from him getting it and send another if not response then in many location you can have the tree removed and send him a bill. In most cases if any part of the tree is on your property you can be responsible for 50% of the bill.
A less legal way is re moves as much of the tree from your property (re weight the tree) if you remove all the limbs on your side of the property line (no permission needed in most places) this reweights the tree so it is heavy away from your property and this way if it does fall it will fall away from your property. If this is the route you decide to go make sure you have the certified letters as this does open you up to some legal liability if the tree falls.