By default, conversations remain open until you close them. Even once you close them, they are never deleted -- they can still be viewed by selecting "View Closed Conversations" under the Filter option in the Messages view:
Auto-Close Conversations option
Under Settings > History > Auto-Close Conversations, you can elect to have any inactive conversations automatically closed after a certain number of days of inactivity. You can specify the number of days. Just like with manually closing conversations, nothing is actually deleted.