Dialogue sometimes reads artificially, dryly, boring, or all those other adjectives that describe something that simply isn’t good.  Heart wrenching dialogue is only heart wrenching if it is written well.  It needs to be sad, and it needs to have pauses, moments to cry, or scream, or whatever your character needs to do.  It needs to basically make your reader want to do all those things, too.

Step 1) Think back to an argument–a serious, maybe break-up argument–and consider the pauses and silences that added intensity to the argument.  This doesn’t only work for arguments (that’s just a good example).  These pauses need to take place within the heart wrenching dialogues that happen in life, such as a loved one dying, a friendship ending, or a last goodbye.
If you want your reader to really feel the situation no matter what it is, break up the dialogue with those pauses without simply saying “He silenced.”  Create your own silence within an argument with a line like, “He stopped talking, stopped moving.  The stillness of the room screamed its silence at me, his words still ringing in the air.”  And go on with something like that depending on how long you want this pause to be.  The overall point of a pause/silence is to reflect the intensity of a real life, emotional situation.

Step 2) Avoid dialogue tags, because in the middle of an emotional conversation, dialogue tags would strip the emotion right off the page.  For more on dialogue tags, check out my article on Dialogue Writing Tips.

Step 3) Be raw.  During heart wrenching dialogues in our lives, we say raw things to each other, and what I mean by that is that we say the things we feel writhing in our chests.  Pry out those words.  If you want emotional dialogue, say what you would be terrified to say in real life.  If the boy you love is about to leave your life forever, tell him you love him.  If you’re cheating on your girlfriend with another girl, tell your girlfriend everything about her that you don’t love.  Get messy.  Get real.  Because it’s raw with feeling.