Well, yes & no!
All of the snotty & hard glue wouldn't have to be removed per say, but it would be best to remove all of the gooey greasy stuff in the floorboards or it will soak into the new carpet over time and get it greasy as well. Also, if you don't remove the snotty glue the new carpet will stick to it and will make it the same nightmare as the original installation if you ever need to lift the carpet up again (for instance if you want to remove it to dry it out from a water leak!!!!).
Though I did go to alot of effort to get all the glue out, in retrospec if I had it all to do again I would just simply stick down the Fatmat directly to the existing glue and not spend the time and effort removing it all. Also, I would not attempt to strip the foam insulation off the tunnel, but instead stick new 1/8" thick Dynaliner to the stock foam (this would eliminate all of the sticky residual glue left on the foam as well as repairing the chunks that are torn off when pulling the old glued carpet off of it). All of the above suggestions would reduce the job to only about 40 hours or so to complete it (about 2 weekends of work total), and would actually still result in as nice of an installation as mine!
Live & learn...