To publish or un-publish your listing, please click on the Profile in the upper right hand corner and from the drop down menu choose My Home.
On the resulting page left hand side, you will see the Listing menu. Scroll down to the Home Publish option and toggle the button to show OFF for un-published listing or ON to publish it.
If you decide to un-publish your listing, your listing won't appear in the search results anymore and it won't be visible to other HomeExchange members. This means that you will not receive any new exchange requests until you decide to publish it again, which you can do at any time ;)