Unfortunately, there's no way to view completed tasks in the iOS device for now. We're hoping to bring this feature in a future update but there's no ETA right now as to when that's going to happen.
You can also vote for it here https://todoist.com/Vote/show/5932/.
How do I view completed items in desktop verison?
There are several ways to do this:
1. Karma/Productivity Trend
On the upper right corner, click the icon beside your Karma points. It should give you several options one of which is "View all completed". Click that and you should get a page of your completed tasks that you can sort by project, date, and label.
2. Completed tasks per project
At the bottom of each project, just below the last task is the icon that resembles a clock within an arrow in the counter-clockwise direction. You can click this to "Show completed tasks" and it should return completed tasks for that project.
If you have any follow-up question, please feel free to ask. :-)