The warrant will remain until it is vacated by the Court, which occurs when the person is arrested. It does not just go away, and the Statute of Limitations does not apply. Good luck.
DISCLAIMER I do not practice law in your State. This answer is provided solely for informational purposes only. This answer does not constitute legal advice, create an attorney-client relationship, or constitute attorney advertising.
Warrants in Washington State are written with an expiration date. As a general rule, the expiration is seven (7) years from issuance but the issuing judge could set an earlier date of expiration. Upon expiration, it would be up to the prosecutor to request that the warrant be re-issued or for the court to do so on its own motion. For DUIs, bench warrants are almost always re-issued upon expiration meaning that the odds of your boyfriend's warrant ever "going away" are very very small.
Also in Washington, on a DUI warrant the DOL will be notified and your boyfriend's driver's license will be suspended until the warrant is addressed and taken out of the system.