I set up this server (http://www.gametracker.com/server_info/64.74.97.151:25200/) and tried posting the banner on the portal page of my website, but the only info the banner displays is "Server not found in the
GameTracker Database". The server is obviously in the database if I can link to it in this post, so why is it showing that error on the site banner?

That would imply that you altered the code on your website to something different than what was given to you on the banners page. Please try recopying it.

Just tried a direct copy/paste from the banner page with the same results. Our 2 other banners work fine, but this one is just being a pain.

You could view the source of the page when it is displayed as "no server found" to see what is not correct in the code. I looked on your site http://www.ironsightgamers.com/ and did not see the issue you describe. If you send me a link to the page where the issue may be seen, I could tell you what is wrong with the code.