You could (and in fact probably should) do what GitHub does. Fetch the images, store a copy and serve them yourself. That way even if the external URLs stop working, the images on your forum/site will remain functional.
This is scary to me so I haven't tried it yet. If I'm allowing users to upload arbitrary files, I might have to deal with things like copyright laws. It doesn't sound fun. Does anyone have relevant information for such a site in the United States?
I'm not a lawyer (nor even American) but I believe the most relevant legislation would be the Digital Millennium Copyright Act. In which case, it might be a good idea to formulate a policy similar to the one linked to in the footer of this very website.
It's not a trivial problem, I'd be more worried about users uploading illegal content maliciously. Would definitely require a moderation team depending on the site size...
AFAIK hosting a link could be copyright violation too. All bittorrent sites host links, not real content, yet they are sued and closed.