Should not doesn't mean does not... As soon as your film starts getting any sort of public viewing (whether just amongst your friends, or on YouTube, or whatever) then you had for sure better start worrying about copyrights. All it takes is one little hint of the song usage to make it back to the copyright owner(s) and they can seriously make your life miserable if they choose to.
Better safe than sorry, besides permissions to use copyrighted materials aren't always that hard to come by. The cost of it all depends on how much viewing the product will receive, so if you're just making a couple copies and showing them around school then it won't necessarily cost that much. (Though, the final cost does depend on who owns the music...) Anyways, its always worth looking into obtaining copyright permissions...
Period.