What is the difference between the following two?
They didn't play the game from their heart.
They didn't play the game from heart.
You didn't say that from your heart
You didn't say that from heart
'Doing something from heart' is not a phrase I have ever heard, and I don't believe it's good English. Doing it from the heart is the usual phrase. Doing it from your heart is uncommon, but probably not rare enough to be unidiomatic; there is no real difference between that and from the heart.
The first idiom, Y Do X From Y's Heart, is OK.
The second one, Y Do X From Heart, is ungrammatical. It should be: Y Do X From The Heart,
They didn't play the game from the heart.
The definite article is part of the idiom, and both idioms mean the same thing.
Whose heart should they play from, after all, if not their own heart(s)?
Maybe there's some idiom here that I'm not familiar with, but to the best of my knowledge, leaving out the "the" is simply wrong. The word "heart" requires an article (or some other subsitute adjective).
To me it sounds like when you're doing something from the heart means you doing it by pity while doing it with the heart shows that you're doing it with passion.