This is a very specific case: When you have a task with a due date visible in Inbox/Inbox section, then open the task, remove the due date and set up a start date in the future, then close the task - the task stays in the Inbox, but when you hover over it, you can see e.g. Starting tomorrow. Reloading the webpage resolves it.
When you open such task and just change the due date to a later one, the Inbox refreshes O.K. and the task disappears properly, the issue is with replacing due date with a start date only.