Many landlords/rental agents run background checks on their potential renters. Therefore, this might come up and affect their decision whether to rent to you or not.
There is nothing inherent in being on bench warrant status which prohibits you from signing a lease.
Why would you want to rent an apartment when you have an open bench warrant? Most people who have those end up with free accommodations from the county. As for the landlord, if he runs a standard background check he may find the warrant. Companies who do these checks are constantly updating their open warrant list.
There are no restrictions based on a warrant but most do credit and or background checks.
This is not a comprehensive answer and it is impossible to provide a meaningful response without a consultation. Call us for more information. 619.797.5456 www.mataelelaw.com
Taking care of the bench warrant is probably a good idea. If you put it on calendar with a court, there's a good chance the judge will set a court date and release you on your own recognizance (depends on some other factors like how long you were evading the court, the charges, etc.). Speak with an attorney first and see what your options are.
If you insist on staying a fugitive, keep in mind that larger complexes with professional management services are more likely to run effective background checks. Your best bet to find an apartment might be in a small, privately-owned unit. Even subletting could work.