I have student's residence permission in Italy, which is valid for 1 year. Now I should move to Germany to stay there for at least 3 months, but according to Schengen rules, I can be there just for this maximum time limit. By the way, I would like to know if it is possible to rent a place without any permission of Germany for this time.
It should be possible but you would basically be looking at vacation rentals. You might have a hard time finding anything on the regular market. Even if you find something, it's not unusual for agent to ask for a fee (Provision) of up to two months rent when signing the contract, which would make the rent much higher if you only stay three months. On top of that, deposit can be up to three months' rent.
I think that intent is also important: If you are on vacation, even for three months, you don't need a permit or registration. If you want to work or make Germany your home, you would need to register and you can't do that with a visa from another country.