Actually, each studio has rules for how and when those trailers can be used and generally they require them to be removed a number of months after the movie premieres.
Now, will they actually come after you if you show their trailers without permission, probably not.
It will be best if you contact them and get their permission. I mean, if you're serious about your movie website you will want to have a contact at the studios for movie trailers, movie stills and other promotion material.
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