There's more to this story... If I had to guess I would say that it's no longer profitable (or at least, very profitable as it once was) and it's not fun to run an image host (imagine all the trouble that comes with just copyright issues, let alone criminal matters). The founder also has a new startup called Pingly which I'm sure is more rewarding to be working on.
On its face Twitpic is shutting down because they won't get their trademark, but they've never had their trademark and have had no problems operating. Twitter isn't suing them, they're just saying they are against the trademark application. They don't need the trademark and it's bizarre to shut your company down for the possibility of being forced to change your name in the future.
If Twitpic wanted to continue running they could without any changes or extra work. It's a dead end though, Twitter has its own image service and there is little reason besides legacy to use Twitpic. Here's their traffic situation which should pretty well mirror revenue as it's web based advertising paying the bills:
https://www.quantcast.com/twitpic.com
They're hemorrhaging traffic. Globally there were 85,043,992 visits in the month period a year ago compared to 13,315,016 in the most recent month. That's also just the past year, Twitpic has been on the decline for years and peaked in July 2011 with a cool 280,021,248 visits.
Expenses are likely not down nearly the same amount as they have to store more and more images every month (with less traffic there is less bandwidth, but this doesn't count images not loaded on their website).
Just in the last month they lost 46% compared to the previous period. It's falling off a cliff. This seems like a good way to get some sympathy on the way out. A rude move to their users though. Maybe Twitter can step in and offer a bottom dollar buyout so that the links don't break.