While potentially more efficient than bogosort for larger input size, this sorting algorithm has a serious limitation. I am of course talking about being limited to 140 characters per tweet. This seriously restricts maximum input size you can sort, which in turn severely cuts down on potential applications of this technology. Moreover, without deployed SAAS (sorting as a service) bot, algorithm is not deterministic which will complicate handling logic as you need to account for being forever alone without anyone to sort your numbers.
In short, I would advise against deploying this on production until technology is more mature.
Merging of tweet shards is also possible via the same mechanism. Just tweeterate through the retrieved shards and merge them one by one. To merge two lists, compare elements pairwise. To compare elements pairwise, construct a two element list and sort this list via the short-form tweetsort api.
For sorting big lists of numbers mergesort could be used, only resorting to twittersort when the list of numbers is small enough [1]. In fact, that's the way mergesort is generally implemented (except the "cutoff" algorithm is usually insertionsort, not tweetsort :p).
The trouble is that some mobile ISPs compress images (as I found to my surprise). I know it should be sacrosanct, but you can't rely on the same bytes coming out the other end. I think OCR would be more robust.
Otherwise you're potentially ruling out mobile users.
Put your numbers in a pastebin, then post a link to the pastebin. Then, when you get the sorted numbers, post the sorted sequence in the original pastebin, thus creating a cloud-backed repository of sorted variants of number sequences. After that, you can check if your sequence is already on pastebin before asking twitter!
In short, I would advise against deploying this on production until technology is more mature.