If a visitor leaves my store and then returns, will the promotion banner still be displayed?
Yes - if the Magento session is still active.
If the visitor uses the same tagged link to return to the store, whether they see the promotion banner again will depend on the combination of 'Promotion lifetime' and 'First time visitor only' settings which you configure:
Note that first time visitor is tracked via cookie information. If the visitor deletes their cookie information, they will count as a new visitor and will see the promotion banner with a new timer countdown (if configured) when they leave and later return to your store.