The biggest struggle with posting ASCII art anywhere was always the formatting. Trying to get your work to use a fixed width font so it doesn’t come out all warped looking. Also, making sure you added carriage returns at the end of each line of text. If you missed one you would soon find out if you posted it somewhere outside of the Notepad file you created the art in.
I don’t know how the Twitter art is posted. That may be as complicated as making the art itself. A lot of the Twitter art seems to be design versus a picture/ image. The designs are nice. But, as an ASCII artist from the olden days, I am biased to love artists who use text to make images and manage to get them to post.
For a quick look at the most recent text art on Twitter search for the hash tags:
tw1tt3rart on Twitter
T_witterArt on Twitter
ASCII_Art on Twitter – Not really ASCII Art but Twitter art and text art.
Daniel Rehn has a Twitter Art list.
TwingDings for Twitter
TwitClipArt – Twitter Drawings Library
Squidoo: One Line Art for Twitter – A good start to practice with. Try othersites to find more one line art or begin creating your own.