I am also experiencing this issue. It also is affecting my Google Drive uploads, but in a different way.
With Dropbox, I have the issue of it showing an insanely high upload rate and only 1 second remaining for the upload to complete (which it never does). Then it completely kills my Internet. When I delete the "uploaded" file from Dropbox, and kill the process, my Internet returns to normal. I can then re-open Dropbox and as long as I am not uploading, it is fine. Syncing downloads from other computers (work, tablets, etc...) seems to be working fine. I just cannot upload from my home computer (which is running Avast).
Google Drive will not upload files. I just get a message to view unsyncable file(s). However, it does not seem to kill my Internet connection like Dropbox does. The files will just not upload.
I disabled all Avast shields and both Dropbox and Google Drive uploaded normally with no effect on my Internet. Turn them back on, and the problem resumes. I have since read that the Web Shield is apparently the one causing the issues. Here is what I am using:
Avast! 2014.9.0.2016
Dropbox 2.6.27
Windows 7 Ultimate
Hopefully an update/fix can address this issue soon. I really like Avast over other antivirus programs that I have used in the past!