Instagram direct messages will stay forever unless you set a photo/video to 'view once' or 'allow replay'.
On the other hand, Instagram photo/video messages disappear after one or two views or remain in the chat, depending on its settings.
Unlike Snapchat, the message is not automatically deleted whenever you send a message on Instagram.
However, if you post a photo or video, you can set it to "view once" or "allow replay", which means the person can only view the photo/video once or twice.