How do I know if my taxes were "approved" if I'm not receiving a refund?

The IRS can question your tax return for three years from the due date or the date that you filed it, whichever is later. Even if you got a refund and they paid the refund, they can still question it later. So there's no final approval until the three year period has ended.

Ok ... some terms need to be explained ... you file a return and it is either Rejected or Accepted for processing.  No return is Approved  but if you have a refund the refund is approved for dispersal.  Both returns with refunds and balance dues  can be reviewed and/or audited usually within a 3 year period but sometimes they can review/audit further back.   So all you can check on with a return with a balance due is the Acceptance when you efile ... you can log in to see if the return was accepted  and/or check your email confirmations of filing and then acceptance.