In various scriptures, it is noted that our soul resides in our heart. What is the significance of soul residing in our heart? Why residing in the heart?
I think it is referring to an emotional connection rather than an intellectual one. This is also the reason why Bhakti is emphasised so much because you can go on analysing, it has no end. Also, this might give some insights:
...Head understands head and heart understands heart. Nose can only smell; eye can only see; ears can only hear, ears cannot see. So, heart can feel, isn’t it? We try to put the heart in the head and the head in the heart and it doesn’t seem to work. Your heart feels something is beautiful, head says it is beautiful. We cling on to the word in the mind, we don’t feel it. We repeat the word ‘beautiful’ and it doesn’t seem beautiful.
The same thing goes for love. You talk too much about love, you get stuck in the head and it doesn’t rise in the heart. In silence love emits, radiates. We are not separate. We are all one. I know you all and when you go deep inside you, you know me also. We are not strangers...
I have also at some other places, references to using your heart being made in the sense of using your intuition. For more, read this article by Sri Sri Ravi Shankar.
“The soul is the core of your being. It is eternal. It doesn’t exist in space/time. It’s a field of infinite possibilities, infinite creativity. It’s your internal reference point with which you should always be in touch.” — Deepak Chopra
In Srila Prabhupada’s purport to Bhagavad-gita 2.17, this issue is addressed:
“The influence of the atomic soul can be spread all over a particular body. According to the Mundaka Upanishad, this atomic soul is situated in the heart of every living entity, and because the measurement of the atomic soul is beyond the power of appreciation of the material scientists, some of them assert foolishly that there is no soul. The individual atomic soul is definitely there in the heart along with the Supersoul, and thus all the energies of bodily movement are emanating from this part of the body. The corpuscles, which carry the oxygen from the lungs, gather energy from the soul. When the soul passes away from this position, the activity of the blood, generating fusion, ceases. Medical science accepts the importance of the red corpuscles, but it cannot ascertain that the source of the energy is the soul. Medical science, however, does admit that the heart is the seat of all energies of the body.”