After I write this I'll try completely bypassing the router, and hooking directly to the modem, and that should definitively say whether or not the router is the cause of the problem.
The ports that are forwarded are being forwarded to this machine (the one that cannot connect), and there aren't any
ports forwarded to the computer that doesn't have connection problems. I've also gone a step further and tested running this machine in the router's DMZ, also to no avail.
It is the router. I bypassed it and connected to different servers and networks with no problem. I've got one last-ditch idea, and I'm going to restore my router to factory settings, and then update its firmware.
Well, I reset the router to factory settings, updated firmware, and reenabled the ports. I don't know what exactly changed, but I can use IRC chat now. Thanks for the help though.