First, none of the countries involved (UK, Australia, Italy) prohibit multiple citizenship today. So if you continue with your plan, your boyfriend will not have to renounce anything.
As mentioned in the comments above, according to Italian nationality law there does not seem to be any automatic acquisition of Italian citizenship upon marriage (although, historically, "Foreign women who married an Italian citizen before 27 April 1983 were automatically granted Italian citizenship.")
So, from your question it sounds like he intends to naturalise as an Australian citizen. That should be no problem. After you have been married for a few years (depending on where you live and whether you have children), he could apply for Italian citizenship through marriage. However, he would only need to do this if that's something you want for your family, there's no requirement for him to do so.
In general, as long as each country permits multiple citizenship, there's no difference between holding two or three or even more citizenships. The only things one might be mindful of are any applicable obligations of citizenship (eg. taxation, military service, voting, and so on).