What is the role of the dialogue editor in volume automation and noise reduction in movie post production?
It depends on the agreements made between the dialogue editor/supervisor and the sound supervisor. Mixers could prefer to not have volume automation, but this differs. Why do you ask? Are you editing dialogue? Or are you mixing the movie?
Recommended reading: Dialogue-Editing-Motion-Pictures
I usually have my Dial editors only do very basic level adjustments. Basically lower any really loud pfx or a line that is dramatically out of sorts with the rest of the scene. With noise reduction, I never want my editor to do any broadband NR or notch filtering. You can get into a lot of unnecessary processing that might not be needed once everything else is in the mix. The only thing I ask my editors to do is deliver me a track that is smooth so when I apply NR I'm dealing with the broader picture.
Several things a dialog editor can and should do with level automation...
I don't discourage editors from using something like iZotope RX on big problem clips, but the original edited clips must remain in sync and muted on an adjacent track. Some editors have become very adept at this, but it must be instantly undoable when mixing.
KellCole is correct when saying that usually the least treatment to dialog is best, and you can't know until all the FX and music are together with the dialog in the mix. So it's best for the editors to err on the side of keeping the work simple and straightforward.
Another thing about Noise:
When I do Dialogue Editing I always search the recordings for pieces of short atmosphere if no extra atmosphere was recorded. Then I save them and deliver them in a folder for each scenery. So the re-recording mixer has a gap-filler if needed.
IMHO its better to stick to the natural noise then remove it.