A justice of the peace will be performing the ceremony for us. Do we need to invite him to the reception?
Probably not. The word "probably" comes into play because the officiant has a particularly unusual role; the authority he or she brings is what actually legitimizes the marriage. So it might feel abrupt to dismiss him immediately after the service. Consider inviting your officiant to the cocktail hour, especially if you spent much time working with him in preparing the service. That invite can be issued verbally.
However, in a case where the justice of the peace has no additional responsibilities and is not previously known to you, you are not obligated to invite him -- and he is most likely not expecting an invitation.
You may want to extend an invitation to the rehearsal dinner if your officiant will help you rehearse. (Just don't be surprised if he declines.) Justices of the peace who charge for their time at a rehearsal, however, needn't be included.