Fed Chair Jerome Powell has tersely indicated that he won't resign if asked by the President and that he cannot be fired or demoted. Peter Conti-Brown, who is a keen observer of Fed issues, has a nice summary of the state of the law. The basic idea is that as a formal legal matter, members of the Federal Reserve Board are removable by the president "for cause," such as inefficiency, neglect of office, and malfeasance. It's less clear if the President can demote the Chair to a mere Board member and, if so, what would be the standard for demotion, but it would seem strange that the greater cannot include the lesser. 

But here's the thing:  none of the legal rules matter. 

Realistically, if President Trump were to fire Powell for any reason, no matter how ridiculous (e.g., "I don't like his tie"), the effect would be to render Powell's position untenable. After a purported dismissal, it would be unclear whether Powell would have authority for any act he would take as Fed Chair. That uncertainty would be crippling; the Fed would not be able to conduct its business as usual if Powell's authority were in question. Remember, Powell isn't just voting on the FOMC or on rules, but he also acts as the executive director of the Fed for its day to day operations. Yes, the issue could ultimately be resolved by the courts, but that would take some, and the Fed can't afford to have its operations hamstrung for weeks or months. 

Now it's possible that Powell could get a TRO, but who would represent him in the litigation? It couldn't be the Fed. He'd have to fund his own representation (which he certainly can), but if he isn't ready to seek a TRO on day 1, it's too late. That's a key lesson from the dispute about who had authority to run the CFPB. And who's to say that Powell could even get a TRO if Trump gave a seemingly plausible reason for dismissal?  If there's a contest here, the President wins.