The date the task is created in the destination project will be captured. For all intents and purposes, this is the form submission date. You can see it at the bottom of the Task Detail pane for each task:
Asana created this task. Apr 22
And if you hover the date it will show it in full in a tooltip.
Asana only shows it there although it is a field in the database. Other tools let you see it more easily, for example Asana2Go (disclaimer: I’m the creator):
Larry