From a performance standpoint, having data in memory is orders of magnitudes faster than having it in a database. There are two reasons to put data into a database instead of storing it in memory:
- You have more data than fits into memory
- You want to make sure no data is lost in case of a (intentional or unintentional) server shutdown
As you already said, poker can be quite fast. To keep response times low, handling each game in memory would be beneficial. However, you might want to estimate how many games you intend to handle simultaneously and how much data you will have per game. When you realize that this will occupy more RAM than your server(s) will have, persisting the whole game state to the database after each move and retrieving it again when the next player makes their move might be necessary. I didn't do the math (I can't, because I don't know your requirements) but eyeballing it I would find it unlikely that this would be required.
However, I would recommend to persist the current state to the database as soon as a round is over, so the chip balance of each player is persisted and no progress is lost in case of a sudden server crash. Having to rollback the current round which wasn't even decided yet would likely be acceptable for most players, but rolling back a high-stakes game several rounds in could make those who already won a lot quite angry.