What is the ruling on touching a ritually impure or dirty phone that displays religious materials, and is the screen considered a barrier? And what is the ruling on using dirty or ritually impure headphones to listen to religious materials?

It is forbidden to touch a phone with something impure if the Holy Quran screen is displayed on it, due to the contradiction with reverence and to observe the difference of opinion regarding the ruling on the electronic Mushaf.

As for listening to the Holy Quran or religious lessons with headphones that have impurity on them, there is no harm in that, because the impurity does not transfer to what is being heard.

If one intends to disrespect the Quran or insult it by any act, that would be strictly forbidden and could even amount to disbelief.