Since this request was made, rules were added to Asana (Business and Enterprise Plans) so you can track the number of due date changes by starting with a rule trigger of Task Updated > Due Date.
A good action to use would be Create new subtask. That will leave residue of each date change which you can then count at any time. Just one subtask may mean the date was set initially (as an early post her pointed out) depending on how you initially get the date set and whether it triggers this rule or not, so you may want to subtract one from the count of these added subtasks to see how many actual date changes there were.
A refinement: If the project already has a Count numeric custom field that subtasks inherit, or if you multihome the subtask to a project with a Count numeric custom field, you can set that value to 1. (You can do all that in the rule action setup step once.) Then if you showed the Count field as a column in list view, it could sum the total for you and you have a counter showing how many times the date was changed for a task, although you’d have to expand the subtasks to see that.
Hope that helps,
Larry