I know Davy Jones from Pirates of the Caribbean cut out his heart because he loved Calypso and after everything she did to him his pain was too much. But why does he not want his heart around him/on his boat?
The Flying Dutchman seems to be the safest place for his heart or at least safer than in the hands of all the other parties.
Also it's no longer in his body so the feelings can't hurt him through the chest, can they?
Actually the heart was kept on Isla Cruces in order for Jones to be able to not feel the pain. This is made clear in PotC : At World's End : In an early scene, Jones is shown playing his organ; after a while he starts crying, then a little later he notices that he is crying. At that point he becomes furious, because the fact that he is crying tells him that the heart has been brought aboard the Dutchman against his wishes (by Cutler Beckett, who has just boarded) - "Get that infernal thing off my ship!".
note: This answer should not be interpreted as excluding the other answers.
It is kinda well-explained in the movie.
But why does he not want his heart around him/ on his boat?
Davy Jones carved out his heart after he betrayed Calypso, the sea goddess. She had left him after this. The reason behind carving the heart out was that he no longer wants to feel the guilt and grief about this betrayal and any kind of affection.
The Flying Dutchman seems to be the safest place for his heart or at least safer than in the hands of all the other parties.
Since anybody who has his heart can make Davy Jones do anything and there can be any mutineer on The Flying Dutchman, he kept it hidden in the chest on the Isla Cruces and the key to that chest within his facial tentacles all the time.
For the same reason it would be kind of pointless to keep your valuables in a safe, then leave the key on top of it. Someone who wants to steal your stuff will only be slightly inconvenienced, rather than prevented from accomplishing the theft.
The key by itself isn't useful to a thief. The chest by itself isn't useful either. The two in close proximity makes the thief's job easy (both on the same ship all the time) and is, therefore, a lot less secure than keeping them apart.
Adding to A J's answer when you say:
The Flying Dutchman seems to be the safest place for his heart or at least safer than in the hands of all the other parties.
Well, in parts you are correct. Imagine if the Flying Dutchman got attacked and invaded, it would be easier for the invaders to find the heart and torture Davy Jones with it. That's why he separated the chest with heart from the key.
