No, conversations do not get deleted from Avochato.
There are four available statuses for conversations in Avochato: New, Pending, Open, and Closed. You can set the status of a conversation by opening it in the Messages view. The status option appear in the upper right hand corner:
If you set a conversation's status to "Closed," it will disappear from the default Messages view. However, you can always view closed conversations by clicking "Filters" in the Messages view and then selecting "Include closed tickets."