My heart is in the top drawer

First off, you spelled "heart" wrong in the title. Just thought you might want to know that.

My heart is in the top drawer
Beating quietly in the corner "Corner" somewhat rhymes with "drawer." As you do not continue with any rhyme scheme, you should change this.
I keep it here for safe keeping This line needs to be redone. It really has no positive effect on the feeling of the poem, and it just generally does not fit with the surrounding lines.
Waiting for someone to brush off the dust

It's worn out, dirty This line could be so much better. So much potential imagery, tossed into the garbage.
It's been torn apart, and glued back together
There are a few scrapes here and there Definitely does not fit as it is. Reword this.
But if it really belongs to you none of that will mater Way too complicated a wording. There are other ways to say "I love you" that are not so confusing.

The beating is continuous, soft Yeah. I think I've gathered that it's continuous. I don't need to be reminded.
If you hold it just right, you can feel it pounding through you Too long.
In time with your own
it's happiest at times like this "This" would be better as "these," and "happiest" is incredibly bland.

I want to give it to you, but i'm nervous "I'm," with a capital I.
You have to promise to be gentile "Gentile" means an individual who does not follow the Jewish faith. I think you mean "gentle."
You promise? Okay, hear it goes
it's in your hands now This stanza needs extreme reworkage. The way you portray your tentativeness is so cliched and unpoetic. The asking of questions and all that.

When you get tired of it, the drawer is still open
What? You say you'll never get tired of holding it Don't need the "what."
No one has ever said that before There are better ways to say this.
So I guess our hearts are beating together now I thought they already were (stanza 3, line 4).

I'm not going to lie, I expected this to suck because you misspelled heart, but I found this simply amazing.

However

I keep it here for safe keeping (this seems awkward, keep it here for safe keeping? TOO MUCH KEEP!)


But if it really belongs to you (should be some punctuation here) none of that will mater



I want to give it to you, but I'm nervous
You have to promise to be gentile
You promise? Okay, hear (HERE) it goes
it's in your hands now