Currently customers can access forms that you no longer want them to through their cache - there's no way to prevent this as the link to the form cannot be changed. Removing the link from the landing page doesn't help if someone has accessed the form previously and then goes back to it after the form's "closing date". Deleting the form (even when cloning it for future use first) means that you can no longer export data which means this is not a way to resolve this.
We're a school admissions department and getting applications in after a rigid closing date causes problems for us.
One solution would be to edit the form in HubSpot, make all of the fields non-required and hidden, and then change the Submit button text to "We are no longer accepting submissions" or whatever is appropriate to your particular CTA.