Assuming, as most questioners related, that the infraction was not a felony, the really short answer is: A bench warrant will fade from being readily available after a few years, but will always be on your record. For a misdemeanor, countless sources and individual accounts online relate how, either personally, or as a story about another person, a bench warrant was outstanding, but did not come to light unless someone strongly pursued a search.
However, in many cases, even the police and the courts do not search very deeply unless they really wants to hold on to a person. License renewals, various traffic violations, little infractions, and the like failed to reveal any immediate notice of an outstanding warrant. On the flip side, several people relate how they were arrested and held on an outstanding warrant. Some situations were due to warrants forgotten in time, when a person was a teenager or young adult. Some situations relate an old warrant causing a problem, even though a case or situation was resolved, the warrant was not cleared or discharged, or closed (whatever the terminology is that jurisdiction). The scariest ones are the ones where a person is arrested on an old warrant; the relationship to that warrant is having the same first and last name and living (or being) in the same state in which the warrant was issued. In these cases, some people lost months of their lives trying to clear their name. There were a few horror stories.
For a felony, the situation is more dramatic. This article is purposely avoiding anything to do with a felony.
So, stipulating that an open warrant never leaves the books unresolved, and a misdemeanor, what should one do about it? Here, it is unfortunate, as several writers noted, that so many people recommend ignoring it. Not having any appreciation for the laws of our land, they recommend that you essentially thumb your nose at the court, and go your merry way. Here, again, lots of people have lots of advice. Amazingly, even lawyers have advice. Most people recommend taking care of it, even if it is out of state. It can come up at a time that will be embarrassing or at a time when someone is doing some kind of background check on you. If it is an out of state warrant, you could even be extradited. These are all good reasons for resolving the situation. Being proactive will say something about you to a judge. As always, retain a lawyer to assist you. Have worthwhile friends and co-workers write crisp letters to your character. Write your own stating the circumstances of the case, the warrant, justify your change in life, how you no longer conduct yourself in a way that led to the infraction. Seek alternatives to traveling to the out of state court. Depending on the infraction leading to the warrant, it is likely not worth the court’s time and cost to pursue extradition, or even having you appear. Contact the district attorney or prosecutor’s office in that state. Ask what might be done to resolve the situation. Work with them. Own up to the warrant and the circumstances.