If you are pulled over in Illinois you will be arrested. Illinois will call the police department in Texas that you are on probation with and notify them. They will then keep you in jail for 72 hours waiting for Texas to drive to Illinois to pick you up. This is unlikely to actually happen. Then you will be released. Your warrant will not be cleared, but you will be released.
If you are arrested and put in jail for an alleged crime while you are in Illinois, they will also call Texas and tell them they have you. Texas will again be called and not wish to drive 300 miles for someone who was arrested for $57. They will have done this when you were arrested so you will not be kept any longer. However, if you were arrested without trial, go to your court date, and are then said to be released, they will keep you in jail up to 72 hours longer waiting for Texas to not come.

