Getting a motion to quash and recall the warrant on file would likely assist in avoiding denial of the visa based on the warrant. You do not have to be here for an attorney to file the motion, but will have to get back to appear eventually.
Normally a warrant for failing to appear in court can be handled by an attorney, but it is ultimately up to the judge to allow a warrant to be quashed without the client even coming to court.
Really depends upon the judge. There is a chance the warrant can be recalled without your being there. At some point during the case, you will be required to appear to answer the DUI charge.
It's not clear if it's an arrest warrant or a warrant for a failure to appear at a court appearance date on the DUI matter. It sounds more like the latter. if you failed to appear at a scheduled court date and the judge issued a warrant - it will be difficult to quash and recall that warrant because it would be a violation of your bail bond to leave the State of Illinois without the Court's permission (which is usually done with a written court order). If it's a warrant for your arrest because the officers obtained it without apprehending you and they convinced a Judge to issue the arrest warrant based on facts supporting your arrest, you can try to have an attorney explain to the Court your circumstances - but it will be a difficult task - and you will eventually have to appear back in Illinois to take care of this situation.