This was a recent concern raised for me because of this question. The asker provides a link to their EMF file which is awesome and definitely helpful, but unfortunately uses File Dropper to do so. From their about page:
How long are the files kept?
The files are kept forever as long as they are being downloaded. If the files are not downloaded even once within 30 days consecutively they are removed. If you have a premium account your files are never deleted.
So basically after 30 days, that link is useless.
How should we handle these posts?
If it's a question that is pivotal on the external link, I would close it as unclear what you're asking. "Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking." If the question has an answer, there's no need to delete the entire question since someone might come back looking for it. A close vote will put it in the review queue, but that might confuse some who overlook why it should be closed. Instead you could flag it for moderator attention explaining the issue and we can handle it.
If it's an answer that depends on the external link, then it probably shouldn't have been an answer in the first place. I'd strongly consider deleting the answer if it has no value without the link.
How can we prevent this from happening?
Certainly re-uploading any images using SE's image uploader is always preffered. However, since imgur only allows image links, perhaps there is another service we could use for other types of files (PDF, PSD, AI, EMF...). It might be a good idea to set up a Dropbox account for GD (and give the credentials to trusted users?) to preserve these types of uploads.