The dead walk and talk.
Actually, the entire season has a theme of dead people walking around talking up a storm, among many other things. But the thing is, the dead don't always know that they are dead, according to the little boy in the Sixth Sense, and there are dead people are walking around everywhere. One thing to make clear about AHS: dead people are only in this house and only the ones that died there. Which made Addy's spirit very happy because she wasn't trapped in the house. So you almost believe that there is something evil in the Murder House that keeps spirits there, against their will. As opposed to the Sixth Sense, where spirits are still around to fix something and why they need the little boy Cole.
At the Murder House, Tate interacts with the entire Harmon family without them knowing that he is dead. At times, he doesn't seem to know that he is dead - I think he is delusional as a ghost as well - but at times he does. Violet had no idea that she was dead and she seemed unaware of the time she missed in school.
Unlike the movie, in AHS people whom the ghosts appear to and interact with do not
always know they are dead. The ghosts appear as healthy as they ever were - Hayden is as crazy dead as she ever was alive - and can totally do anything a person can do. Tate kills Chad, Hayden has sex with Travis to prove that she still can and the twins, Troy and Bryan pelt Ben with eggs during Halloween.
The open cabinets.
While not all of the ghosts are willing - or able? - to appear to everyone at the house, they do things to get attention. Adelaide was blamed by Vivien for opening all of the cabinets in her kitchen while her back was turned. Very much the same way Cole's mother asks him why he opened all of the cabinets in the kitchen when she walked out of the room and right back again. We see later that in the case of Sixth Sense, its the crazy housewife who is opening the cabinets and Addy tells Constance that the Twins did it.
In both the film and AHS show, teen suicide is used to bring out the horror of early death.Violet kills herself and doesn't realize she is dead. The same is true for the boy who shot himself with his dad's gun in the Sixth Sense. The underlying premise - that kids sometimes die at their own hand - is horrifying.
So far, these are the ideas and themes I see that American Horror Story did to give a nod to the cultural horror classic The Sixth Sense. I'm sure there will be more coming.