I'm trying to get the Apple TV App to work but I don't own any Apple devices

If you don't have any Apple devices you must be using the Apple TV app on a Smart TV?

Have you tried the instructions in the support document Get the Apple TV app or Apple TV+ app on your smart TV, streaming device, or game console?

It seems to give two options to sign in - scanning a QR code on your mobile phone (not necessarily an iPhone) and also using your TV remote control.

Once the Apple TV app is on your Smart TV -

If you're not signed in with your Apple ID, go to Settings, select Accounts, then choose one of the following Sign In options:

• Sign In On Mobile Device: Use your phone's camera to scan the QR code on the TV screen. Then follow the instructions on your mobile device.
• Sign In On This TV: Use your TV remote to manually sign in with your Apple ID and complete setup.

By the way, an iTunes store account uses the same Apple ID you use for every Apple service or hardware device.

If you don’t have an existing Apple ID, then you can create one. It will need your personal information, a validated email address (respond to test message), and a payment method. How to create a new Apple ID - Apple Support

The iTunes account is an Apple ID account that is ready for iTunes Store services, including subscription streaming services such as Apple TV+. If you set up the Apple TV app on your smart TV, you might be redirected to activate.apple.com to create your Apple ID.

Without an Apple mobile device, I don’t think the QR code method will work directly, as that is used as a shortcut to enter your Apple ID credentials without typing. The mobile device would need to be already registered with your Apple ID account, for that to be beneficial. Use manual sign-in instead.