The above post is right. A lot of the faculty just look at the stuff online rather than print it out.
Usually the programs have set their computers to automatically download stuff once/week or so I think. So they should/will get updated LOR's, etc. as long as they are uploaded to the ERAS postoffice. Usually you can check on ERAS (I think it's call the ADTS -applicant document tracking system screen) to see if/when a program has been downloading stuff. Usually they will review whatever you get in before their deadline...I mean in terms of extra letters of recommendation.
For what it's worth, I had at least one program ask me for a fellowship interview even though my application lacked one of the LOR's required by the deadline. Most of the time that doesn't happen, though.