Sometimes. Most runners take care to use the restroom before starting the race. On some occasions, runners have dropped out of the race, temporarily, to use the facilities provided along the race course.
Some few incontinent runners will wear special undergarments to avoid embarrassment, although this is hard to disguise in most professional running apparel.
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Elite runners typically don't; however, if they must, down their legs. For others, there are typically porta-potties every several miles along the course. Males (and some females) often resort to wooded or other secluded areas.