I could download the image and print it myself, someone is making it easier. They don't even look to the image to see if it is copyrighted, porn or just Lorem Ipsum. I get you point but if they are a "we print anything you send our way" shop, how can they be liable for what I send them? I guess you can only blame the end user...
I imagine it would be similar to any site based on user generated content where they would first need to contractually prohibit users from uploading material that infringes on third party intellectual property rights and then they would need to have a policy in place to address any complaints by third party intellectual property rights holders that their right were being infringed. But it certainly wouldn't be a safe bet to just assume their off the hook entirely because it's too hard for them to determine infringement.
In the US, the "making it easier" part can make you liable as well. In fact, a huge amount of copyright-related cases have considered this exact issue. Take a look at all the P2P suits, as an example.
In my uneducated opinion, I believe this would be similar to YouTube. YouTube allows anyone to upload anything, and they aren't required to preview the content before it is publicly available on their site. However, they must comply with DMCA takedown requests.
I think ClickShirt would have to do the same. Anyone can print any image until the owner of the image issues a takedown request.
The added wrinkle is that anything being put into the ClickShirt system is from a different website -- it's hosted somewhere else, not on a user's computer. That's different from YouTube, because it's fair to assume that ClickShirt is ALMOST ALWAYS using copyrighted images.
Regardless, I think they'll get shut down in the US. This isn't the kind of thing that courts are very lenient about. Neat idea, though.
You certainly don't have permission to sell T-shirts with my artwork on them..