For some reason, I suddenly couldn't open links or select text in my iTerm2. After some experimenting, I discovered that it only applied to the tab which contained WeeChat. I could select things normally and open links if I held down alt (or option in Apple speak) simultaneously.
It turns out mouse integration had somehow been enabled in WeeChat. I could click channels to switch to them and swipe left and right to browse channels. Extremely cool, but not at all what I wanted.
Simple disabling mouse integration in WeeChat solved the problem:
Well. Now that I know the feature exists and actually works with my stack at work (comprising WeeChat, tmux, ssh and iTerm2), I wonder if I'll ever find use for it. Probably not. But cool none the less!