I do not know clearly, and each situation may be different. Technically, the answer is no. If you have a work visa and you stop working, the visa needs to be cancelled, and the employer may cancel it.

However, I have had a 1 year work visa in the past, and I left my job after 3 months to return to my home country. I came back to China when the visa still had 2 months left on it, and it worked fine for the 2 months, and I am on a new visa now. The government has never given me any troubles. 

The technical and legal answer is no.  But if the former employer do not notify authorities that you have left, the working and residence permits will remain valid until expiration.  Remember, your work visa is good only for 30 days after entering China, then the work and residence permits take over  and become your visa.

What should happen is than when you resign, contract ends or you are fired, the employer notifies PSB, and they will give you 30 days to either find a new job or if not, leave the country.  If you do find a new job, you have 10 days to change employer's name from your permits.  If not, you will need to get a new work visa at your home country.