Walking a dog is already illegal in Tehran but it looks like they want to ban pets ownership entirely in all of Iran or at the very least impose a $$$ permit requirement issued by a special committee. In other words good luck getting it.
Corporal punishment for dog owners was even discussed at one time.
I am aware of the Muslim tradition of viewing dogs as unpure but this goes beyond that to include all pets to stamp out what they view as a symbol of Westernization.
The country where having a pet could soon land you in jail
https://www.bbc.com/news/world-middle-east-62205744