My HSA paid for the full dental charge. My insurance covered a portion and sent me a check. Do I need to report the insurance refund check as income?

If you don't correct it, then yes it becomes income.  Worse still, it is not only subject to income tax but also a 20% tax penalty for a non-medical withdrawal of your HSA.  The reason why is that an HSA may be used tax-free to pay for unreimbursed medical expenses.  Once you received a refund check from your insurance, the amount of your refund is a reimbursement.  Since it is attributed to your HSA distribution, this would be the treatment on your distribution.

But you can fix this before April 17.  If you have the amounts of the insurance reimbursements, you may return these to your HSA via your plan administrator as mistaken distributions.  They are mistaken because you believed you were covering an unreimbursed medical expense.  So if you return the amount to the HSA before April 17 (the filing deadline), you will be able to correct this.  

Here's a website with more information on this:  http://www.hsaforamerica.com/hsablog/how-to-fix-the-most-common-hsa-mistake/

"Correcting" in this case means contacting the HSA custodian and reporting a "mistaken distribution" (as Daniel said). The mistaken distribution is the amount of the insurance check (i.e., the part of the bill that should not have been paid from the HSA).

What the custodian will do is ask you to sign a form and send them the check for the mistaken distribution. Be nice to them when you ask, because they don't have to accept this request. You should have this money, because it's the insurance check.

Once this is done, the custodian paperwork will properly report your situation.

NOTE: do this before the due date of the return - either April 15, 2020 or October 15, 2020 if you have filed an extension.