What I normally do is let him open presents in front of the gift givers that give him things in person and save things that are mailed or left for him to give him throughout the year. I always unwrap everything so we can send thank you notes, and put the others in my closet for a rainy day, long car trip, etc. If it's a duplicate of or similar to something he already has, we donate it.
I think most 2 year olds would be overwhelmed after 2 or 3 presents at a time.