I've seen libraries take pictures of book covers and promote them on their library social media page, and was wondering if the same policy holds for movies. Can we take a picture of the front covers our new DVDs and promote them on Facebook? Or is it preferred that patrons browse our new DVDs in the library and/or on our library catalog?
Even though we purchased a movie license, I do know that due to copyright laws, we are not allowed to promote on social media any movies that we are showing at the library. Patrons are asked to call us and inquire what movie we are showing. Thanks for the help!
There are a lot of legal technicalities hidden in this question, but before we get to them, here is my overall advice: The more your library generates unique, custom content showing the people, personalities and experiences of your library on your social media, the more you can include copyright and trademark-restricted content in social media promotions.
For example: If a library takes a picture of a new DVD, with only the cover content in the photo, and posts it to social media, that could potentially trigger some type of copyright/trademark concern. But if that same library takes a picture of their librarian holding that same DVD while giving a thumbs-up ("This new movie is librarian-approved!") that concern is greatly reduced, since the proprietary work is only part of the message.
This same guidance applies for book covers, new games, and other media packaging. Since copyright and/or trademark can both be invoked to protect any of that content (although just how protected the content is will vary from item to item), displaying it on social media as part of your library's overall personality and outreach is much better than using a photo or scan of the book/cover on its own. When you don't have permission, use of a proprietary image as part of a related but larger social media message ("We have this great book!") is generally a safer approach to image use.
Now, at this point I must note: the "image use" in this question is different from using cover content (let's call them "thumbnails," for nostalgia's sake) supplied by providers like Overdrive for your library's online catalog. Use of thumbnail content in your catalog (and thus, generally, on your library's website) is likely restricted in the license from the provider, but supplied with the understanding that the thumbnails will be viewed via your library's website as part of the service.
Now, as to announcing movie nights...this question gave me a double-take, because neither copyright nor trademark, in and of themselves, bar listing the bald fact that your library is hosting a (licensed) movie night, and the title of the movie—whether via a poster, or via social media.
But since I have never known a librarian to submit a baseless question to this service, I dug a bit more, and found this statement in the Swank guidance for libraries using their "single event' license:
"If the public library’s social media accounts are set to private, the title may be used. If the library’s social media accounts are not set to private, it is recommended the title not be included. The movie event may be promoted on the social media pages with a link to the title on the library’s website."
So to be clear: copyright doesn't forbid promoting the movie, but restrictions on promotion could be a requirement of the license (the contract allowing your library to show the movie), or (as the case here) a "recommendation" of the licensor, likely at the request of the trademark holder. This is one of the more bizarre "recommendations" I have run into in the contract-analysis business, and I thank the member for sending it along!
And that's it. Again, the take-away from this answer is: the more your library generates unique, custom content showing the people, personalities and experiences of your library on your social media, the more you can include copyright and trademark-restricted content in social media promotions as a component of that larger messaging. Along with being a type of risk management, this will also lend itself to your library having better, richer, and more interesting social media, so it's a win-win; sometimes the law and quality control go hand-in-hand.
And now, to go watch the new [TITLE OF MOVIE REDACTED] with my family.
 For purposes of this question, I do not consider your library's website "social media," since in many ways these days the website is simply the virtual part of the library. "Social media," to me, are third-party contractors: the usual and growing array, like FB, IG, TT, Twitter, etc.
 Using movies stills and original posters can pose a concern, but here, we're just talking about announcing the title.
 You can find it, too: https://www.swank.com/public-libraries/faq/#afterpurchase6
 That said: "it is recommended" is not the hallmark of contractually enforceable language. My guess is that this is something Swank told its content providers it would do, but everyone realized that as a hard requirement, it is pretty ridiculous ("We're showing a movie! Can't say the title!") and could cost them business. I can see why content providers would ask for it, though, and I bet it shows up in other licenses. If you have a license with a requirement like this, please send us a copy; I collect contract artifacts like this.
 Because it will make it much easier to claim fair use, and also make it much less likely that your library will be accused of infringement in the first place.