The glass negatives that Burton used were gelatin dry plate negatives. 'Glass negative' can also refer to wet plate collodion negatives (I believe Chris is referring to these when he says there are many who still pour and shoot glass plates.) As opposed to collodion photography, you don't need much in the way of special equipment for gelatin dry plates. You will need a plate holder, but they are easily made from old wooden film holders.
Originally Posted by BetterSense
Enlarging for printing is identical as with film negatives, with the flatness benefit you pointed out. My 4x5 plates fit perfectly in my 4x5 negative carrier. I run into trouble with 5x7 because my holder is a glass carrier a half inch larger in both dimensions. I have set a 5x7 plate on the glass like I would a negative and the glass cover sheet. Works fine except for perhaps just a hint of light bounce.
There's more info on dry plate photography here:
Hope you give it a try!