Short answer
It is permissible, but many Hanafi muftis advise against it. A lock screen with an ayah goes everywhere you go, into the bathroom, onto the floor and into hands without wudu, so the risk of disrespect is high. The same concern applies to Allah's name; a plain design is the safer choice.
Using an ayah as a wallpaper is not haram in itself, but it is disliked because it exposes the words of Allah to places and situations that do not suit them. If you use one, cover or turn off the screen in the bathroom and avoid laying the phone face up on the floor.
The four schools agree on this.
Explanation
Honouring the symbols of Allah comes from the piety of hearts (22:32). The Hanafi texts already dislike writing the Quran on walls and prayer niches, out of fear that the writing will fall or be treated without care. A phone wallpaper is a smaller version of the same concern, with more risk: it lights up in toilets, crowded trains and noisy places, sits on the floor, and is handled all day without wudu. The four schools share this concern for respect.
If you want the reminder, there are better options: set a reminder notification with the ayah's meaning, keep a Quran app on your home screen, or use a widget that you can switch off. AmalKit's ayah of the day gives you a daily verse without keeping it on your lock screen. On taking a phone into the bathroom, see reciting in the bathroom.
Framed calligraphy at home is different, because it stays in one respectful place. Hang it above head height, keep it out of bathrooms, and avoid hanging it where people sit with their backs or feet toward it. On ayat as ringtones, see playing the Quran in the background.
The evidence
Quran, Surah Al-Hajj 22:32
ذَٰلِكَ وَمَن يُعَظِّمْ شَعَٰٓئِرَ ٱللَّهِ فَإِنَّهَا مِن تَقْوَى ٱلْقُلُوبِ ﴿٣٢﴾
That [is so]. And whoever honors the symbols of Allah - indeed, it is from the piety of hearts.
Respect for Allah's words guides where they should be displayed.
Where the ruling is recorded
- al-Fatawa al-Hindiyya, Kitab al-Karahiyya, chapter on the manners of the mushaf and writing the Quran. Writing the Quran on walls and prayer niches is not recommended, for fear of the writing falling or being disrespected.
- Radd al-Muhtar (Ibn Abidin), Kitab al-Taharah, discussion of writing the Quran on walls and objects. The same concern for respect.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.