What will/should the arbiter do if the side which cannot theoretically lose resigns? Assume that in an endgame, one side cannot theoretically lose (for example, his opponent has only a king) and he is not in any time crisis. In this case, what will/should an arbiter do if this player resigns? Will/should the arbiter accept the result of the resignation, or will/should the arbiter consider this as bad sportsmanship and take further actions against this player?
Though almost no reasonable player will do this under normal circumstances, I am still wondering if this had ever happened in professional games before.
The first thing to note is that the FIDE Laws of Chess are silent on this. Resignation at any time in any situation is allowed and immediately ends the game:

5.1.2 The game is won by the player whose opponent declares he resigns. This immediately ends the game.

Other methods of losing are qualified in some way. For example, checkmate is only valid if the preceding move was legal, a game where one player runs out of time is only a loss if there is some way (i.e. helpmate) that the opponent can deliver checkmate.
However this is not the end of the story. There are also the Anti-Cheating Regulations. They say:


These regulations deal with the investigation of suspected cheating incidents.
“Cheating” in these regulations means:
i) the deliberate use of electronic devices (Art. 11.3.2 FIDE Laws of Chess) or other sources &gt; of information or advice (Art. 11.3.1 FIDE Laws of Chess) during a
game; or
ii) the manipulation of chess competitions such as,
including but not limited to, result manipulation, sandbagging, match fixing, rating fraud, false identity, and deliberate participation in
fictitious tournaments or games.


If the arbiter believes the resignation was a result of ineptitude then it should stand. There have been examples of top GMs resigning drawn positions through ignorance and ineptitude.
If the arbiter suspects some form of cheating or receives a complaint of such then this should be investigated. These things also happen.