The consequences of turning yourself in for a traffic warrant are going to vary depending on the situation. In some rare instances, you may find yourself going to jail, since you do after all have a warrant for your arrest. More commonly, however, you will have to resolve the underlying issue associated with your warrant and pay additional fees associated with that warrant.
For example, if a warrant was issued for your arrest because you failed to show up in court after receiving a traffic citation and summons to appear for a hearing, you will generally have to pay all fines associated with the original ticket you received, in addition to court costs and fees associated with the warrant.
By not showing up in court on your initial court date, you generally lose the right to defend yourself or to plead not guilty to the initial citation you received. This means you will not have a way out of the situation other than to pay your fine. If you are unable to do so, the court in some cases may allow you to do community service in lieu of paying a fine.
In many states an "amnesty date" is offered periodically. This is a day on which you can come to court in order to deal with your traffic warrant and resolve the underlying issue without fear of arrest.
Another way to make sure you avoid being arrested for turning yourself in for a traffic warrant is to hire a lawyer. Your attorney can negotiate an agreement or deal on your behalf whereby you resolve all outstanding claims against you.