To contribute to an HSA, you must have High Deductible Health Plan coverage (HDHP). Your HR department should know if this is the case. Then the employer usually helps you open an HSA, although this HSA belongs to YOU, not your employer.
It sounds like you have a Health FSA (ask your HR Department), and not an HSA.
Yes, if this is true, you can delete the 8889. However, that may not always be sufficient.
If the HSA stuff keeps coming back, you should do an HSA RESET to get rid of all HSA data, even some you are not aware of.
***Reset***
1. make a copy of your W-2(s) (if you don't have the paper copies)
2. delete your W-2(s) (use the garbage can icon next to the W-2(s) on the Income screen)
*** Desktop***
3. go to View (at the top), choose Forms, and select the desired form(s) 1099-SA (if one), 8889-T, 8889-S (if one), and 8853 p1 (if one). Note the Delete Form button at the bottom of the screen.
*** Online ***
3. go to Tax Tools (on the left), and navigate to Tools->Delete a form
4. delete form(s) 1099-SA (if one), 8889-T, 8889-S (if one), and 8853 p1 (if one).
5. go back and re-add your W-2(s), preferably adding them manually
6. Now continue with your return.
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