Hi, Chat Room plugin developer here.
No, currently old messages are not deleted automatically. I intend to add that in at some point (I should have more time over Xmas to do that).
The messages don't take up a huge amount of room but in the meantime you can delete old messages by running a query like this:
DELETE FROM qa_chat_posts WHERE postid<1234
Replace 1234 with a recent postid e.g. 100 less than the highest one in the database, so if the latest message has postid 12300, use 12200 in the query.