I just got an extension for my J-1 visa. This means, that I got another DS-2019 form. Since I didn't leave the country, I did not get an updated visa stamp in my passport. Now I want to renew my DL and the DMV asks me to provide them with my visa and my DS-2019. Obviously, I don't have a valid visa stamp. Will they still accept my expired visa stamp?
According to https://www.dmv.ca.gov/portal/dmv/detail/dl/dl_info#BDLP, the proof of birth date and "legal presence" that most likely applies to you is Valid foreign passport with a valid Record of Arrival/Departure (form I-94).
If they want to see your visa and DS-2019, they probably understand that a visa can be expired once you've entered the US. As long as your passport, I-94, and DS-2019 are valid, you should have no problem.
From your comment:
I quote from the letter: "Copies of your Visa, your unexpired passport, your i94 (front and back) and your DS-2019 (valid 60 days or more)" just after quoting this, I feel the visa can be expired. I should've looked more at the details...
You are correct. They ask explicitly for an unexpired passport, but put no such condition on the visa. A copy of your expired visa will therefore be sufficient.
I renewed my driving license with little more than my I-797 and I'm on an H1b.
My original visa has expired and new one granted, but since I haven't left the US I don't have the actual visa in my passport.
This was no issue for the GA drivers license dept.
Check the web site of the licensing dept of the state you're in for a list of documentation you require since different states may have different rules.