How long does the DA have to prosecute me for burglary?

If you are being accused of a first degree burglary then you are facing up to  six years in state prison and a strike offense.
If you are facing second degree burglary you are facing up to three years in jail.

In either case the DA must file formal charges against you within three years of the date of the alleged crime. However, remember they do not have to arrest you within that period of time. They merely have to file a formal complaint against you and file it with the court within three years of the date of the alleged crime.