There are a few different solutions for this out there, especially clients that run some custom protocol between the GUI and the bouncer, like Quassel and Smuxi - but I've tried both, and each both (a) has poor mobile support and (b) once you get used to it, turns out to just suck in general. (These opinions are a few years out of date, though.)
Thankfully I don't actually care much about mobile support: I don't use IRC for time-sensitive discussions, and for recreational purposes, trying to participate in a real-time conversation where everyone can type several times faster than you is, IMO, a pain; I'm faster at typing on iOS than I used to be, but it's still just not comparable to a real keyboard. Better to save the chatting for when I have one. But I find ssh+screen an unacceptable solution: partly because of issues with notifications (i.e. out of the box, there are none) and copy+paste, though both can be fixed in theory, but mainly because of the latency. When the letters I type don't appear for 100ms or more, it really trips me up and I make a lot more typos. I could find a server with less latency at home, but that wouldn't help when traveling, especially on an unreliable connection - even though there is no fundamental reason chat should be latency sensitive in the slightest. Tried mosh (ssh + prediction) as a compromise for a while, and it works better, but it has some issues and doesn't fully hide the latency.
These days I'm using Glowing Bear, the web-based remote for weechat (that uses yet another custom protocol). Latency's gone, and like other web-based clients it has the neat feature of embedding YouTube videos and images, so I'm finally pretty satisfied with my IRC setup. (And I can be paranoid about security and run it on localhost rather than using their website directly.) But there are disadvantages: Glowing Bear is somewhat feature poor, it (again like other web-based clients) is relatively slow to render, and there is no native iOS weechat remote. (Guess I could still use weechat as a regular bouncer, or perhaps try Glowing Bear from mobile Safari...)