Create a refresh shortcut
Changing dates, postponing or completing tasks, etc leave a lot of things on the list that no longer need to be there. Todoist should have a shortcut key (such as "r") to force a fresh of the page similar to the way Toodledo does it.
In the mini version of Todoist (used by all plugins), you can click on the reload icon at the bottom. On the full web-app, you can use F5 to refresh the page or click on the Todoist logo.