I think something like this is a wonderful gesture from one friend to another. Paying off a person's mortgage means saving that person A LOT of money, especially considering the interest the person saves. I don't see where this would be harmful at all, and I would be personally eternally grateful.
Ed Sheeran helped his friend in an unique way. He was able to allow her to work for something, rather than just giving her charity. It helps her because she will make some money that will continue as the song continues to sell and play. It was a great thing with no long-term harm.
The situation is entirely helpful in nature, and Sheeran did not place any further obligation on her. This gives little reason for trouble to arise from this incident. Of course it could be the case that she receives negative publicity due to being involved with someone famous, but she instead has mainly progressed in the music industry. Negative publicity usually has little effect anyway.
No, paying off a friend's mortgage does not do more long-term harm to the recipient. The recipient has a home that is mortgage free. this allows the recipient of Ed Sheeran's gift to save money for other things and have more disposable income than the person would without this gift.