Stay permit

I got my asylum in Italy but never worked there. Currently I'm working in a different country. Can I still renew my permisso di soggiorno in Italy.

No you cant , if you already left italy they and have a permit in another eu contry they now you have that and will revoke it . If you are in that other country illegaly , you can , but if the authorities of that country catch you , your status in italy will be revoked and they will send you back home . Dont play with this.