How do I get a refund for an unwanted Duolingo subscription on iPhone?

DuoLingo offers subscriptions both on their website where it is managed by them and with an in-app purchase where it is managed by Apple. For the in-app purchase managed by Apple, you would cancel that subscription here:

View or cancel app subscriptions on iPhone - Apple Support

It is always your responsibility to cancel a free trial at least 24 hours before the renewal period. I always suggest setting a Reminder to notify you to cancel a free trial. None of apps that offer a free trial I have used sent me a reminder to cancel a free trial. When you double click the side button with both the free trial period and the subscription price on the screen, you are authorizing being billed for the subscription if the trial is not cancelled at least a day before.

In most cases all transactions in the App Store are final due to the explicit actions needed to approve the purchase. At times Apple may provide a compassionate refund, but it should never be expected. To request a refund for an Apple managed purchase, that is done here.

Request a refund for apps or content that you bought from Apple - Apple Support

Since you said you had received an invoice, there is a link you can tap in that email to request a refund.

You have 5 options, but I don't see any clear path in getting a refund:

• Write a review for the app in the App Store. This will alert other potential users and it will allow Duolingo to provide a response for other users to see. This will hold them accountable.
• Contact Duolingo like you did. They cannot issue you a refund through the App Store, but nothing stops them from sending you a free promo code for a future Duolingo subscription or even sending you money directly. I doubt that would happen, but they are the ones that made the promise to you.
• Report Duolingo to Apple as a scam/fraud since they mislead you. When you sign in at reportaproblem.apple.com, instead of choosing Refund, there is an option to Report as Scam/Fraud.
• Dispute the charge with your payment method. I would not recommend this approach as it will likely result in your Apple Account being disabled. Your agreement with Apple was to pay for the subscription if not cancelled at least 24 hours before the trial period ended, so you may find that it is no longer possible to use the App Store at all.
• Contact Apple directly. At least you will be able to state your case to a person instead of the automated refund process. Contact Apple Support - Apple Support