Asana's behavior regarding entry of dates can be a little confusing. Asana does not let you enter just the "start date" portion and leave the "due date" portion blank. If you do that, Asana transfers the "start date" you entered to be the "due date" and blanks out the "start date" - and worse, it doesn't warn you that it did that!
To avoid this problem when you're wanting to enter a start date, make sure you enter both a "start date" and "due date". Furthermore, the dates you enter for start and due date must be two different dates- if you enter the same date for both start and due dates, Asana does the same thing as described above - it uses that date as the due date and blanks out the start date!
Please also see this article for more details.