If it's something that connects outwards from PythonAnywhere (as opposed to running a server that accepts incoming network connections) then it should be fine. IRC bots are normally outward-connecting things.
Probably the best thing to do is to upgrade and try it -- if it doesn't work you can always downgrade again, and we have a 30-day no-quibble money-back guarantee.
One thing to look out for -- when you upgrade to a paid account, your old consoles will still be running in the "free" mode, so you'll need to restart them. And if you have a web app, it will also need reloading.
Also -- as your bot will presumably be running as a long-running process, you should probably check out this page on how to write programs that keep running even if our console servers are reset for maintenance.