It’s kind of buried in the two links I provided. Basically the creation of the District of Columbia by plan laid out the residence of the POTUS as the WH. The District of Columbia was created so that the POTUS’s residence (along with the rest or the government) was not at the whim of some state.
3 U.S.C § 102:
The President shall receive in full for his services during the term for which he shall have been elected compensation in the aggregate amount of $400,000 a year, to be paid monthly, and in addition an expense allowance of $50,000 to assist in defraying expenses relating to or resulting from the discharge of his official duties. Any unused amount of such expense allowance shall revert to the Treasury pursuant to section 1552 of title 31, United States Code. No amount of such expense allowance shall be included in the gross income of the President. He shall be entitled also to the use of the furniture and other effects belonging to the United States and kept in the Executive Residence at the White House.
The president doesn’t have a lease on the White House and he is not a tenant.
3 U.S. Code § 102 says, in relevant part, “[The president] shall be entitled also to the use of the furniture and other effects belonging to the United States and kept in the Executive Residence at the White House.” His getting to use it means that he gets to keep other people, like the old guy who had the job, from using it. This is a very specific provision of law unique to the president and, in general, any time there is a specific provision of law that conflicts with a more general provision, such as general laws on eviction, the more specific provision controls so the president gets to occupy the White House and gets to kick out whoever else he wants. He has at his disposal scads of federal law enforcement officers to enforce that right and they will all benefit from
It’s a weird fantasy though that the Secret Service is going to have to bodily remove the ex-president. If Trump isn’t president anymore, he will move out. It doesn’t matter if he hangs around the building too long because it isn’t being in the White House that makes him president. I sat in the Oval Office once. I could not command the Seventh Fleet.
As for why the president works in the White House and not elsewhere, the truth is that he does work elsewhere. This lame duck president works, among other places, at his boarding house in Florida and some second-rate golf course in New Jersey. He works in the White House primarily because the dozens people who work most closely with him have offices there. The Old Executive Office Building next door has hundreds of more offices for more staff. The Department of Treasury has its headquarters a few feet away and the Office of Management and Budget is across the street. The people that the president needs to run the government are all in close proximity to the White House so it makes sense to work there. It doesn’t have to be that way. The president could work essentially anywhere but those staffers aren’t going to magically get new offices in Florida. There is no room in the budget to just move them somewhere else.