I found three Scriptures that came as close as I could get to this concept. You'll notice that not a single one mentions heart... or any other internal organ either, for that matter.
John 14:20 On that day you will know that I am in my Father, and you in me, and I in you.
John 17:22-23 ... so that they may be one, as we are one, I in them and you in me, that they may become completely one, so that the world may know that you have sent me and have loved them even as you have loved me.
1 Corinthians 6:19 Or do you not know that your body is a temple of the Holy Spirit within you, which you have from God...
I then wrote the following. (OK, so it's sorta stream of consciousness. Roll with that bit, eh?)
Saying that "Jesus is in your heart" seems to compartmentalize him, even constrict him. It puts Jesus in a part of us. And he sits there inactively. There is no mention of him working through us, of him drawing us to be more like him, discipling us, etc. It's a completed sort of statement It's a location rather than a state of being or a position of operation.
Now consider the similar statement that one needs to "ask Jesus into their heart." This implies that a finite creature can actually direct the actions of an (the) infinite creator. In my recent studying of the book of John I see numerous references to belief and that Jesus came that we might believe. No where do I see that he came so that we might put him in our hearts.
Add to this the prayer of Jesus in John 17. He asked that we might be in him and him in God so that we may be completely one, so that 1) the world would know God sent Jesus and 2) the world would know God loves us. It seems to me that the purpose of Christ in us is in no way tied to our salvation (as "asking Jesus into our heart" would have us believe) but is specifically that the world would have a better understanding of God.
Today people use the "Jesus in my heart" thing as a dividing line. There's us -- we have Jesus safely tucked away inside our blood pumping organ. And there's them -- they don't have Jesus anywhere in them. Not even in their livers or kidneys. Rather than focusing on the unity that John 17's "they may be one as we are one" should point us toward, we first of all twist the concept of Jesus in us and then use it as a sword point to poke others with.
The moral of the story? Christians are dorks. (Which is another way of describing the doctrine of sin, I suppose.)