When we were looking for van #2 (even van #1 for that matter), we had to eliminate even the low roof sprinter because I think they are just over 8' tall.
Here is a photo of Dookie ('84 E350 4x4 w/ SMB poptop) on 315/75/16's sneaking under our 8' garage door:
I'd say that as long as you don't have an elaborate rack (ie, rails only without towers) that a typical E350 4x4 with 265 or 285 tires should slide under an 8' door. A UJOR 6" van on 315's might be pushing it.
Our Express 1500 AWD with CCV top also just slides in - I think it measures 7'-11" at the tallest point. This constraint will be saving money since a lift and upsized tires are out of the question!