Can I stay in Thailand until my Non-B Visa expires after quitting my teaching job?

you are fine, as long you did not get the work permit and the extension of stay you can stay.

The difference between a Non Immigrant Visa and the Extension of Stay.

Not only a visual difference, see picture.

For the VISA, top picture:

The Admitted until ../../.. period you got on entry with the visa stays valid if you stop working. NO, it can not be cancelled by angry employers or the Ministry of Labor. Even when you have the 1 year Multiple Entry Non Immigrant B visa you can keep on using it till the valid until date mentioned on the visa.

For the Extension of Stay based on working inside Thailand. (bottom picture)

YES, this has to be cancelled on your last day of working. You can do this in advance with documents from your employer and the Ministry of Labor where you cancelled your Work Permit.

If you wait till the last day, you have to leave the country immediately and to prevent this, you could apply for a new Extension of Stay and if not approved you will get a big stamp in your passport and 7 days to pack the bags and leave the country.

I removed a list of comments, who obvious do not know the difference between a visa and Extension of Stay.

If you have a Non-B visa (the sticker) that is NOT expired, you can stay in Thailand until the visa's expiration date. IF you have an extension of stay (the stamp), then the extension becomes invalided the day you quit your job.

@Tod ********

...or Robert Lagas may soon be of helpful assist to you.

Where are you in Thailand...what immigration office?

Did you have a work permit for teaching? Your NonB visa for teaching...was this your first year and or was it an extension?