Can I make money off a website about a TV show?

Assuming you have permission to use the copyrighted material from the TV show on your website, you can probably sell ads to support your site. However, the copyright owner may choose to put some restrictions on what kind of ads you sell and how often they appear.

There are many issues involved in setting up a tribute/fan/critic website to a show. As long as those side issues are handled properly, you should have no problem running advertisements on the site.

You should get a lawyer to work with you on this. You can run a site that gives reviews and analysis. And many shows provide the media with clips and photos to use for such purposes. You might also have Fair Use to use short clips that you are using for review or analysis purposes. What you cannot do is use photos anyone else has taken, unless you get a written contract with them first. However, you might be able to take stills off the video clips and use stills provided by the company. I do this kind of legal work and I think you need a lawyer.