Once the sun dies and the solar system goes dark, would Europa (if it had life) continue to provide a liquid ocean and a biosphere?

Yes. The answer is yes. Europa get’s energy by the gravity and closeness to Jupiter which tugs and pulls at it as it goes around.
Therefore, whatever the sun does, this force will continue because Jupiter will continue to orbit the dead sun.
There are forms of life on Earth, deep under the ocean, that are fueled by the heat energy of under-ocean volcanic activity. They derive no energy from the sun due to their depth in the sea. This suggests life can exist without deriving energy from the sun.
This suggests that volcanic activity exists without the sun, and that’s doubtful. Earth’s outlook isn’t positive, but Jupiter will be there and Jupiter will have gravity.
The same could and should be true on Europa.