Both the BitPay and Copay wallets can sweep a paper wallet. To do so, press Send at the bottom of your wallet.
From there, either scan the private key with your camera or copy the private key into the address field.
After you either copy or scan your private key, you should see a screen that looks like this.
Now press Sweep paper wallet and you will be able to sweep your paper wallet into the BitPay or Copay wallet.