Yes, missions do expire from your profile and are automatically released to the public pool if not completed during the allotted timeframe.
How do I view the expiration hours of a mission?
Each mission has a certain number of hours in which you must complete the mission. For example, a mission may have expiration hours of 36. This means that you have 36 hours after claiming the mission to complete it.
1. On the "available missions" screen, click on a mission that you are interested in.
2. You will find the expiration hours for the mission here:
3. To see how much time you have left to complete a mission, go to your "my missions" screen. You will then see how many hours remain here: