There is a story in the Bible that Adam and Eve ate the fruit and thus sinned against God. But other religions, including some Christians, describe that they were having intercourse and that's why God threw them out of the garden. Which Christian groups believe this, and what is their reasoning?
I believe that you are referring to the theory that the original sin was sex. This, however, is not Biblically true. Before the fall, God said:
And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth.
How could they multiply if sex was a sin? The original sin, according to the Bible, was disobedience. (See Genesis 3)
EDIT: Because the question was edited, my answer has now been edited too. I have ran into a few people, mostly from the Catholic faith, who believed in this teaching. As far as I know, this is not Catholic doctrine, they just had been misinformed.
I am not aware of any group who believes this. Adam and Eve's sin is regarding disobedience. The only other teaching (which is false) about anything in connection to the events that unfolded in The Garden of Eden was the first mate (first wife) belief. Sex itself what produces life. This is between mates, for if it was sin, it would not make sense for the earth to be populated. It only becomes sin if brazen conduct and immorality is involved, then you have a problem. But nothing of this sort took place in The Garden of Eden, despite Adam and Eve being kicked out of the garden, moreover, they were already a couple.