I have a pending tax dispute in Chicago and I am worried about traveling to Dubai for a holiday this summer. My lawyer in the States says it is just a civil matter for now but I am afraid it might turn into a criminal warrant while I am abroad. Does the Dubai police check for USA warrants at the airport during immigration? Can I be detained for a non-violent financial problem from another country? What are the chances of being sent back to the USA if I am just a tourist there for a week?

Yes, there is definitely a risk if the USA puts your name in the Interpol database even as a warning. Even if it starts as a civil matter, they can flip it to "tax evasion" which is a crime in almost every country. Dubai is very high-tech and they scan every passport against international lists at the border. You can find more details on the website https://extraditionlawyers.ae/ about how these "private" warrants work in the UAE. I had a cousin who was stopped at the airport over a commercial dispute from New York. He had to hire a local firm to get bail and it was a total nightmare for the family. My advice is to check your status through a professional before you fly. The UAE takes US requests very seriously even without a formal treaty because they want to keep a good relationship. It is better to spend a bit on a search than to spend weeks in a detention center while they figure out the legal details. Just be proactive and don't take any chances with international travel until you are sure.