No. Only contact with a police offer will cause you to be arrested. The DMV will not know if the warrant is an arrest warrant so it is highly unlikely that they will contact the police to have you arrested.
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Mr. Orr is correct. Only police officers arrest people for warrants, not DMV.
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As the other attorneys have said, you should be okay in this scenario. However, you many want to have a backup plan ready just in case things don't go as planned. Speak with an experienced DUI attorney ahead of time and have their contact info with you when you go to the DMV. If you happen to be arrested (either there or any other time before or after), then you'll have a plan ready for representation and taking care of the warrant ASAP. Use this website to find some attorneys in your area and take the time to call them too.
I agree with the other responses, but there are three things I wanted to add. First, if you have an outstanding judgement or conditions from a DUI probationary sentence then you will not be able to apply for a DL without clearing those up first. Second, if you are able to retain a DL I would have bail money on me at all time while driving just in case I was pulled over for a broken tail light or any other small infraction. Third, when you do decide to take care of the warrant, contact a DUI attorney beforehand, and they can arrange for you to turn yourself into the Court instead of the police which will probably save you from ever going into custody again. Good luck.