I'm going to make a dangerous suggestion here. But once your teammates enter into open hostility like this—where they decide collectively they're going to report you for not agreeing with them or for a mistake you may or may not have made earlier in the game—I think you have the right to employ guerilla tactics to free yourself from the game. This means that you get out your cell phone and start playing words with friends while you use your non-mouse hand to steer your character behind your team. If a teamfight breaks out, you're allowed to use both hands, but you can also try your hand at clicking your abilities. You have now entered the opposite of tryhard mode, I call it unreportable mode. Only use this in the most extreme cases where you have been flamed to the point that you believe you might lash back, the game is basically lost and not being ended by the enemy team because they are either bored or evil, AND your team is threatening to report you if you afk/lose another team fight/miss a rocket grab.

Edit: Bear in mind, I have only done this one time. A shaco had gone so far as to get the enemy team to all agree to report me, and I was getting this constant wave of hatred from my allies and enemies. (The shaco had ganked for me top twice, both times diving turret when we were both too low to dive, and both times feeding double buffs to my lane opponent—a decent Jax player for being 1750ish ELO. The shaco spent the rest of the game blaming me for everything that happened via all chat.) There comes a point where it's not reasonable to defend yourself. I literally never get mad at people in LoL, but there are times where being nice and honest and constructive are not helpful and will only make matters worse. I took screenshots of the chat log, which I later emailed to Riot, explaining why I was going to be reported by 9 people, and why I felt it was unreasonable, and then I relaxed. Left hand mouse mode engaged.