It really varies tremendously by chef. Some just sell their name for a fee, and have nothing to do with the restaurant except attending the opening night party. Others run it as their own business, even if they are there only 1 day per month.

Alain Ducasse is a great example of the latter, and maybe a reason why he has earned the most Michelin stars of any chef in the world -- far more than any of the chefs mentioned on this thread so far, and maybe as many as all of them combined (19 by last count). At one point Ducasse had 3 Michelin star rated restaurants in 3 different cities -- all being run hands-on. So, it is possible, but doesn't allow for TV shows. Ducasse doesn't do airline food -- but does do the menu planning for the European Space Agency (through ADF). And runs 2 cooking schools, which shows where his priorities, and heart, lies.