If I allow users to upload media (images, videos, etc.) and I get a DMCA takedown notice for a specific file, do I have to prevent users from uploading that same file in the future?
[standard "talk to a real lawyer for a real answer" disclaimer]
I don't know that it's specifically line-itemed as a requirement, but if the current Flava Works vs. Gunter suit is anything to go by, not making some attempt at this is probably not going to help your case. Note that in the first paragraph of the Court Case section, the judge threw out all of the plaintiffs claims except for the bit where Flava Works basically did nothing to prevent/filter repeat offenses. From the filing:
Plaintiff does not allege that defendants failed to remove the material from myVidster, but does allege that after the notices were sent, the website "continued to be updated with more and more infringing material from its members." (First Am. Compl. ¶¶ 54-57.) According to plaintiff, myVidster "does not have in place any filters or identifiers to prevent" its copyrighted material from being re-posted by repeat infringers, and it "took no action toward stopping, reprimanding, or banning" repeat infringers.
Also remember that this is what YouTube eventually ended up having to build, as ContentID, which goes a step even further than preventing "files" from being uploaded; it actually analyzes the content to try and identify things to reject. (This is obviously way on the far end of options, but it's an interesting example.)