Here is the bottom line on this problem of white around the hearts.  There is a certain softness around the hearts that have a more transparent edge.  That softness picks up any background color you choose, and it also picks up some white.  It's because of the softness.  You can't get a clean file with any softness on the edges.  But it is the softness that gets rid of the jaggies.  So you have to make a choice here.  If you don't mind the slight jaggy edge, and you don't know what kind of background your image is going on, then do it like above.  Make those deletions after expanding.

What if you do know the color your gif file is going on?  Then use it as the background color of your file.  Now you can soften up the edge more so it is really smooth.  Do both the heart and the image layer.  Click on the background and expand the selection by either 2 or 3 pixels.  Your choice.   While the selection is expanded, click on the Gaussian Blur.  Setting of 1.