Big Bump for this.
It happens to me at a lesser extent. I can actually get into the game, however the firewall prompt always comes up first.
This has been happening ever since the Varus patch (no issues before then).
I've also Google'd ways to fix it but nothing has worked (changing firewall/antivirus settings, port forwarding, repairing through the Launcher etc.)
The one thing you can do is wait for the first firewall error prompt, wait about 7 seconds and then click the Retry button. This avoids the second firewall error coming up. However the fact that it even comes up once (delays you about 30 seconds to the loading screen) is an annoyance.
Recently (since Rengar patch), it has gotten even worse in that the whole screen goes black after champ select and alt-tabbing is not possible. The first firewall prompt still comes up after 20-40 seconds, but until it actually gets to the loading screen, the computer is stuck with a black screen.
Before (Varus Patch -> Rengar Patch), it remained at the LoL splash screen, which would appear above anything else you had open, covering up a tiny amount of space in the middle of the screen. However you could still use the computer until the prompt came up.
Some wrenchmen have tried to help (in other threads) but understandably, they haven't been able to offer anything beyond general connection based advice.
It would be great if a Red could reply to this and acknowledge that this is a problem they are aware / not aware of (whatever the case is).
EDIT: A screenshot showing the Firewall Prompt I've been talking about.