Yes, if you fail to pay a seat belt or other moving violation citation that you received in California your driver's license can be suspended by the CA Department of Motor Vehicles once they are notified by the court that you failed to appear (in court) or pay the fine.
According to the California Courts site when you sign your ticket, you promise to go to court or pay the fine. If you do not go to court or pay your fine, your driver's license can be suspended by the CA DMV and you may not be able to renew your car's registration.
The judge can also charge you with "contempt of court" or "failure to pay a fine." If that happens, the court can charge you with a misdemeanor and issue a warrant for your arrest or add an additional fee (called a "civil assessment") of up to $300.
The Superior Court of California for the County of Shasta notes that if you have received a notice from the Department of Motor Vehicles indicating a Failure to Appear (FTA) hold on your driver's license, you may clear it by posting and forfeiting the entire bail amount OR by appearing in court. The hold on your license will be electronically removed from your record within 48 hours of your payment or your court appearance.
If you have a Failure to Pay (FTP) hold on your driver's license, the hold will be released upon payment in full by verified funds (cash, money order, cashier's check, personal check). Please note that a payment by personal check must clear your bank before DMV will be notified to release the hold. This process may take 30 to 45 days depending on your bank.
The county court for the jurisdiction in which you received your seat belt citation should be able to give you information on how they specifically deal with seat belt tickets and those who do not pay them by the date on the citation.
A seat belt ticket in CA would normally be cited as a violation of Section 27315 of the Vehicle Code. Here it states that this infraction is punishable by a fine of not more than twenty dollars ($20) for a first offense, and a fine of not more than fifty dollars ($50) for each subsequent offense. In lieu of the fine and any penalty assessment or court costs, the court may order that a person convicted of a first offense attend a school for traffic violators or any other court-approved program in which the proper use of safety belts is demonstrated.
Since the fine for this offense would appear to be $50 or less it would seem that paying the fine or requesting traffic school would be much better than ignoring the ticket and facing not only a higher fine amount but a possible suspension of your driver's license and not being able to renew your car's registration.
To read about the penalties one may face if you fail to appear in court or pay the fine after receiving a citation you can read sections of the CA vehicle code such as 40508, 40509, 40509.1 and 40509.5. To the best of our knowledge your driver's license could be suspended by the California DMV until the ticket and failure to pay issue is resolved however you should contact the CA DMV to get specific information.