Yes, you are using someone else's property without permission. You are using the trademarks of the sports team in question. While you might be able to get away with using it yourself, sharing it will probably get you, at a minimum, a cease-and-desist letter from the lawyers for that team. By asking your question, you are showing that you know it isn't right.
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If you merely do this for yourself, it will not be actionable. More specifically, for your own personal use the courts would likely consider that de minimus and sports orgs do not go after fans for creating such things for themselves (e.g., making their own shirts, hats, signs, etc.).
Sharing it online and making it available to others crosses the line and would be a dangerous proposition even if you are giving away at no charge.
Please consult a lawyer in private before jumping in here.
DISCLAIMER: this is not intended to be specific legal advice and should not be relied upon as such. No attorney-client relationship is formed with the law firm of Natoli-Lapin, LLC on the basis of this posting.
Q:"Can I create a desktop wallpaper using a sports teams logo and share it on the internet?"
A: You can until a Judge orders you to stop. You would need to be ready to pay up to $150,000 in statutory damages plus the attorney fees of the sports team when they sue you. Perhaps they don't care. If it's just on your desktop and not shared on the Internet, then that is likely to be risk free. When you go out on the Internet, that will be a major problem, as that will be trademark and copyright infringement and will appear to the leagues in their monitoring service reports.
Q: "Does this infringe on copyright laws or anything?"
A: Yes, and more trademark infringement laws, trademark dilution laws, unfair competition laws, anti-fraud laws, and others both state and Federal.
So, on your own desktop for your own private use and enjoyment and nowhere else - Fine. Anywhere else - illegal and dangerous.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.