As far as other Schengen countries are concerned, yes, any D visa or residence permit issued by another Schengen country authorizes the bearer to travel within the Schengen area, subject to the 90/180 rule, unless there is a specific annotation to the contrary.
I suppose it is possible that Italian law imposes a travel restriction, requiring you to stay inside Italy. Some countries have similar restrictions in various circumstances. I think this is unlikely, but I do not know the relevant Italian law. In any event, that question is better suited to Expatriates, since it's about your relationship with your country of temporary residence.
As far as traveling to other countries in the Schengen area, which is on topic here, you have nothing to worry about.