Sports team logos are trademarks in addition to being protected under copyright law.
So even if your proposed use of the logos was "fair" under copyright law -- which is very, very doubtful -- you could still not use the logos as you propose because that would infringe the various team's trademark rights in their logos. And, in that situation, the trademark "fair use" doctrine [as opposed to the copyright "fair use" doctrine] would certainly not apply because you would be using the logos as they were intended as trademarks [rather than descriptively as the trademark fair use doctrine requires].
In short, you can't lawfully do what you propose. But speak to your own intellectual property attorney because there may be facts in play that you're not disclosing or don't know are important.
If you run an ad supported website, it's commerical and it profits you, so don't think there's a difference, from a trademark owner's perspective, between "direct" profits and your interpretation of "indirect" profits based on the source of the profits.
While copyright use does look at whether a user's use is commercial, as opposed to academic/educational, TM use looks to whether the user's use nominatively refers to someone's TM, and uses it as little as possible to get the reference, or whether the user uses the TM as a TM, to denote their brand, and competes with the owner.
Your proposed use is pretty clearly not fair. You might want to search Avvo for all the questions that mention the NFL, MLB, NBA, NHL, NCAA, etc. to see the various queries from people who would like to find a way ti use these TMs. Generally, the proposed uses are TM violations, but occasionally the uses seem legit.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
I think more information is necessary to make an accurate assessment of what you're doing. Your only statement about the use of the professional teams' names or logos is that the users of the website could select which is their favorite. You don't provide any information about the actual purpose of the website or anything additional which would give an idea if it is fair use or not. There are complex legal issues at play and the information provided is insufficient to give you solid feedback.
Use of the logos would be subject to copyright violations, and the commercial nature of your website, as noted by my colleagues, would tip the scale towards this not being fair use.
Use of the names alone could be a different story depending on how they're being used on the website. The names alone would fall under trademark law. (Note that the logos would also fall under trademark as well, in addition to the copyright protection.) If you are asking the users to select their favorite team for purposes of data collection or to group them in some manner, then that is fine. There is nothing wrong with surveying people for their favorite NFL team because you're using the trademarks in a way to identify the actual goods. You're not using them in some manner which is likely to result in confusion as to the source of the goods (this is the question asked when determining infringement of a trademark).
You really need to discuss this with an attorney who specializes in trademark law before moving forward or halting the project. There are facts that you haven't mentioned in your questions which could substantially change things.
Disclaimer: In no way is this information considered a legal advice, but is only a statement of the law. This information does not form a client-attorney relationship.