As unbelievable as it may sound, there is now a Twitter client for the C64. It is called Breadbox64 and it was written by Johan Van den Brande. It is built on top of the Contiki network stack and requires a RR-Net compatible Ethernet card.
I have sent my first tweet from my C64 and it was good.
If you don't believe me, see for yourself. Here is a video from the author of the software: