I totally understand why (and actually thankful that) there has never been a Live Action attempt at a Legion of Superheroes movie or series.
Making their first appearance in the late 1950's, The Legion of Superheroes predates The Fantastic Four, The Teen Titans and The X-Men, making it the first fully-teen-aged superhero group and also the first to present members with a wide range of powers and backgrounds. Having members from different worlds also makes it the first "ethnicity-flexible" team ever.
So how come we've had two versions of the FF and a dozen X-Movies and not one of the first ever teen-aged superhero team in history? Listed below are the obvious reasons why and my unsolicited answers to them:
Problem 1: Movie Technology
The technology to pull off a superhero team film set in the 30th century with a multitude of alien species and worlds (since Earth had already made contact with other solar systems) replete with robots, flying cars, spacecrafts, gadgets and other elements that would make the future, well, futuristic.
With today's movie technology, however, and as seen in Guardians of the Galaxy, Star Wars and Star Trek, as well as other films set in the far-flung future, it can now be possible. Coruscant, Mos Eisley, Xandar, the Star Trek Federation Earth, are all examples of how a galaxy-spanning future Earth would be like, and would be easily created with CGI.
Problem 2: Superpowers
Okay, I admit this is very much related to Problem One, having to do with the technology to pull it off. Until a few years ago, it was almost impossible to imagine how to do The Flash realistically, but now we see it done EVERY WEEK IN A TV SHOW.
And the Legion's powers have been more or less done before - Cosmic Boy's magnetism, Saturn Girl's psychic powers, Lightning Lad's electricity, and so on, so one only needs to think of a better and more creative way of presenting them.
Oh, and for those who are now thinking "So they've been done before, so why do it again?" remember that this story is set in the 30th century, so the modes of transportation, the structures, the fashion, and mostly everything else is different from where Magneto and Electro and jean Grey use their powers. This makes for a more creative venue for the use of the said powers. It's like comparing a car chase from The Fast and the Furious with the flying speeder chase in Star Wars Episode 2. Same chase, different vehicles, way different results.
Problem 3: Empathy
This, in my humble opinion, is why futuristic superhero stories often fail. Not only are the worlds unfamiliar, but the characters and their problems are also hard to relate to. Its easier to empathize with the alienation felt by The X-Men than with the identity problems of a half-human, half-Vulcan, or with the attempts of Data (from Star Trek TNG) to become human.
The Legion of Superheroes are a bunch of teens who live in a future where superheros are no longer needed. But a new threat forces the first three founding members - Cosmic Boy, Saturn Girl and Lightning Lad - to join forces, rekindling the heroic spirit by becoming the galaxy's new set of superheroes.
THAT is the whole storyline right there. Imagine a world/universe so peaceful that superheroes are no longer needed, until a new kind of threat appears and a band of outcasts and angst-ridden teens step up to save the day. That's already the X-Men and Guardians of the Galaxy combined. The tagline could be "Heroes Exist In Every Century" or something like that.
Then you have the colorful characters with a 30th century twist, like Karate Kid who studies a centuries old martial art that, like superheroes, have been rendered obsolete by the Utopian society. Brainiac 5, the latest descendant of Superman's nemesis of the same name, who must prove that he is not like his ancestor (like the son of a gangster who must prove that he isn't a criminal). Ferro Lad, whose deformity forces him to wear a mask, because no matter what century you are in, people still judge other people by their external appearance. And a dozen or more other storylines that today's youth can relate to.
Make it funny, make it exciting, make it about teens from the future having the same problems as teens in the present. Make mine The Legion!