Italy never entering the war would have all kinds of consequences besides possibly preserving the Italian monarchy. No Italian assault on Greece, therefore no reason for Germany to go through Yugoslavia to pull their chestnuts out of the fire. No involvement of Yugoslavia, therefore no temptation to Bulgaria to pitch in to at last get their hands on Macedonia. Italian Libya separating German forces in French North Africa from British in Egypt, so no desert campaign. No second Great Siege of Malta, in fact the Royal Navy would have been under far less pressure in the Mediterranean generally.
Heck, it could even lead to no Pearl Harbour! The British triumph at Taranto was an inspiration to Japanese naval planners, also fewer assets tied up holding the Mediterranean might have meant more in the Far East to deter Japan. It's a stretch, but not I think a totally ludicrous one. And no Pearl Harbour of course means no or later American entry to the war, prolonging the conflict if not changing its outcome.
But if Mussolini had had the wisdom à la Franco to stay out then yes it probably would have preserved the Italian monarchy, there was no particular reason for it to fall otherwise. Less drastic means of keeping the House of Savoy enthroned would as mentioned be an honest referendum and/or Victor Emmanuel III abdicating earlier in favour of the popular Umberto, as he could and should have done. With a figurehead untarnished by the Fascist regime having had time to establish himself, the referendum might have been won decisively.