The problem isn't with having Avast installed, it's the Web Shields feature. It is NOT my internet connection, as the download works just fine with Web Shields turned off. I've done a bit more testing on this that you have given me credit for. With Web Shields turned on, the Forge installer generates zero traffic to files.minecraftforge.net, and its TCP connections (as shown by Process Explorer) are being redirected to Avast's proxy server, and the download simply fails. It also prevents CodeChickenCore from downloading CodeChickenLib. Note that I have no problems downloading these files with my internet browser, only (unsigned?) Java applications seem to be affected.
Given the names of the files in question, the files from both sources should be the exact same version of the library. However, the files in the zip are short by 500 Kb and 1 Mb, and archiving software detects them as incomplete.
I am not currently having a problem, I've already solved it. I'm just posting the solution for the benefit of everyone else.