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It could be taken down (the producers could file a "DMCA" (Digital Millennium Copyright Act) request to remove copyrighted materials. YouTube does not HAVE to post any videos or creative works, being a private company, and can restrict content from being used. As for being infringing, to the extent that the use is satire, comment, criticism, etc... it would probably be considered "fair use", and therefore not infringing.
In short, fair use is a limitation on copyright, a defense to infringement. Here's the law:
17 U.S.C. § 107Notwithstanding the provisions of sections 17 U.S.C. § 106 and 17 U.S.C. § 106A, the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include:
The fact that a work is unpublished shall not itself bar a finding of fair use if such finding is made upon consideration of all the above factors.
Typically satire falls into criticism or comment, and certainly if it is criticism itself then that could be used.
So I doubt that this could even be infringement in the first place, and I doubt that the producers would spend time and effort trying to sue you personally. But a DMCA request is easy (merely having to contact YouTube, assert that they are the owners and that the use is unauthorized). So while you almost certainly would not get in "trouble", there's a fair to good chance that YouTube would still take it down, depending on how much of the work is used AND whether the producers would find out about the work AND whether they would care to send said DMCA notice.
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To follow up I just want to clarify. The video is 4 min long, for about 15 seconds we list a few of the merchandised items from the show while showing a photo of each of them(Duck Dynasty Lunchbox, Duck Dynasty Underwear, etc..) We then spend the rest of the time talking about items that "should" be added to their line. Each of these products we made up but to give them an "authentic" look we added their picture/logo to our items. Would this still fall under the DMCA?
To the extent that you use copyrighted material (i.e. Logos, pictures, video, etc...) it could easily fall within DMCA.
Again, fair use would likely say that this is not infringement, but because fair use is a defense in a legal action for copyright infringement, that would be beyond the situation where YouTube takes a video down for a DMCA notice.
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