OK, I've made some progress on this. Here's an example of a thread with what I'm talking about:
http://fuckcombustion.com/threads/arizer-portable-heater-tech-discussion.25914/ In the second post,
@Andreaerdna posts an image, but I just get ''. I tried hitting Reply like @blokenoname suggested, and managed to get the image to appear in the reply box. If I open the Inspector on the [IMG] in the post, you see it is actually an image element! There is text there too (with a pseudoelement) but the image tag is there. However, the image has not loaded.
Imgur have blocked FC, as discussed in the link @Summer posted, and if you look at the Network tab in your Inspector you'll see a 403 response. More technical details here: https://webapps.stackexchange.com/questions/25681/how-to-remedy-imgur-403-forbidden-errors
The technology that allows Imgur to do this is called the referer header: it tells a website where you came from. I use Firefox so I installed this https://addons.mozilla.org/en-US/firefox/addon/smart-referer/ and it now loads the images! Better to not use Imgu, but useful for the images that are on there.
As for migration, with DB access someone could write a script to find imgur links, download from imgur, reup somewhere else (why not set up S3 and offer image hosting while you're at it? :) ), and then replace the links