This has happened to me before, it wouldn’t authenticate my account when I tried logging into either HOTS or Overwatch, I dunno which. I think the battlenet app just needed to be restarted.
I have had this problem a couple of times until I realized I had to use the Blizz Authenticator you can download on the App Store or Google Play.
Once you open the app, there should be a “Enter code manually” button. Click that and then a code will be displayed for you to insert into the 'Enter authentication code" box on the Authentication screen when logging in on HOTS.