What mortgage origination date do I put for 1098 if we have no mortgage?

It depends. If you are entering mortgage interest for a loan that has been paid off, then you should enter the date for the loan the mortgage interest belongs to. If not, please provide clarification for your question.

We have not had a mortgage for several years, so there is no interest paid or earned

It won't let me off the page until I put a date in for our mortgage, but we have no mortgage

You will need to delete the 1098-INT that carried forward from last year. Here is how to find it:

1. Select Federal on the left side menu.
2. Under  Deductions and Credits,  find Mortgage Interest under Your Home.
3. Select Start or Update to the right.
4. Look for the entry carried over from last year.
5. Look for a trash can, or Delete to right of that entry. Select the trash can and confirm you want to delete.