No. The states do a pretty good job of connecting their records reporting to one another.
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This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Laws vary widely from state to state. You should rely only on the advice given to you during a personal consultation by a local attorney who is thoroughly familiar with state laws and the area of practice in which your concern lies.
Most states communicate electronically. Generally, if your driving privileges are revoked in one state then before you can regain your privileges in another state you must first be eligible in the state where the violation/revocation occurred. In other words, no, you must finish the classes and pay the fines/reinstatement fees.
Disclaimer: This communication is not intended as legal advice, and no attorney client relationship results. Please consult your own attorney for legal advice. This is for entertainment purposes only.
If you dont pay the fines/complete the classes, you are likely to be brought back into court for a hearing on an Order to Show Cause. The purpose of the hearing would be to determine whether you should be found in violation of your probation. In most first time DUI cases, the defendant is sentenced to 180 days in jail with most of that jail time suspended. If you don't pay the fines or complete the other terms, you run the risk of being found in violation of the probation and having the suspended jail and the suspended portion of the fine imposed. In other words, you should pay the fine and complete the classes or run the risk of a more serious consequence than the loss of your license. If you are ordered to appear for that hearing and you fail to do so, a warrant for your arrest will be issued.
The question of whether you will be able to get a license in other state is a touch more complicated. The previous answers to this question are correct; if Utah decides to suspend your license, you are unlikely to be able to obtain a license elsewhere.
You would be much better off contacting the court and attempting to make alternate arrangements concerning your fine than just letting it go and hoping it doesn't come back to create problems for you. Chances are it will cause problems at some point down the line.