My student visa in Israel expired one week ago, but I'm planning to stay here 2 more months and renew the visa either when I'm back home in Italy in the summer, or re-enter in Israel next year as a tourist and renew it then. Can I do so? Can I stay in Israel even if my student visa has expired? Will there be a problem when I leave the country in August to go back to Italy?
As you wrote your visa is already expired, so currently you are there illegally. I didn't find official statements, but normally you will get fined for each day you overstay and if you plan to leave after 2 months after the expired date sure you will be questioned with the risk of an issue of deportation.
My suggestion is to contact the Italian Embassy and ask their help.
It can depend if you are jewish or not, and which school you go to. Many of my friends and myself have overstayed our A2 student visa but as we are jewish yeshiva students there is not much they can do to us, becouse israels law of return states that any jew can come to israel. A Non jewish student would be another story. Currently my visa is expiered, in previos time swhen i left and returned with a expired visa, they renewed it upon reentry.