Yes, start dates are available in the API with start_on. For example, you could GET the start date of a task with this cURL request:
curl -H "Authorization: Bearer 0/123456789" \ https://app.asana.com/api/1.0/tasks/12345?opt_expand=start_on
If you only need to see start dates using the JSON export, simply add ,start_on to the end of the URL once you’ve done an export to JSON.
For example, after exporting to JSON, the URL should look like this:
https://app.asana.com/api/1.0/projects/123456/tasks?opt_pretty&opt_expand=(this|subtasks%2B)
Just add ,start_on to the end of the URL so it looks like this:
Then hit enter.
We hope to have start dates added to the API documentation soon.