In many cases, a failure to pay or appear is a separate offense. I am not licensed to practice in your state and so can't answer your question with any confidence, you should ask an attorney in your area. It depends if your warrant is for a failure to appear at the hearing, or a new offense of failing to pay or appear for the prior case.
My experience the answer is no, since they really have nothing to do with the other. Warrants have a life of their own, there is no such thing as a Statute of Limitations on a beach warrant. Bases on my experience and your question, it probably means you failed to satisfy one of the conditions of probation.
My suggestion would be to pay the fines, have your attorney file a Motion and set a hearing with the presiding judge to have the warrant withdrawn. As of right now, if you were stopped for any type of traffic infraction, a simple check will display the warrant and you will be arrested - even if it's a mistake. Bottom line, pay the fines and get the warrant withdrawn, for it will not automatically go away.
Best of luck
Typically, once a warrant issues you or your lawyer has to appear and ask for it to be recalled. It is unlikely that mailing a check will prompt the clerk to notify the judge to recall the warrant. The easiest way to find out the status of the warrant is to call the clerk and ask. If there is a warrant out, you need to clear it up ASAP. If you hire a lawyer, they may be able to clear it up for you.
If there is a bench warrant it is probably not for simply failure to pay fines. Maybe the case got reset to Court for report to find out why you have not paid the fines or the case was set for a restitution hearing and you did not attend. Consult with a traffic ticket attorney in your area to help you resolve the issue.