It goes both ways.
Fans get too involved and base their happiness on the success of a sports team. Crazy if you ask me. If you're happiness is dependent upon sports, you got no life. These type of fans think the next coach has all the answers and we're gonna be dominant with him in charge (see Meat Chicken with Harbaugh as the top example I can think of).
A LOT of Coaches on the other hand think if I can get this next better job, I'll win more and be more successful, make more $$$$, and be happier (the list of examples here is longer than my arm). One of the few coaches I've ever heard who has it right was Villanova's Jay Wright who said, "If you're happy don't mess with it." The guy after some time won a National Title too.