If you have not a Refugee status, you must have a valid “Permesso di soggiorno” (stay permit) that specifically allows you to work. If you have refugee status, you can start a business.
You must prove that you have enough money available to open and manage your business.
More information
https://www.unhcr.it/cosa-facciamo/soluzioni-durevoli/iohodiritto
You can ask for more details to the nearest One-stop shop (SUAP) office.
E.g. see https://www.milomb.camcom.it/cittadini-extracomunitari