Okey-dokey. I think I understand what you want to do. I'm not very good at explaining things but I hope this helps a little!
What you need to do is cut out the part you want to animate. The easiest way to do this is to cut/paste sections of your drawing. The first five tools in gimp are probably what you'll want to use. The rectangle, ellipse, free, fuzzy and colour select tools (hover over the tool icons to see their names and a description) will enable you to select certain parts of your drawing (each tool selects in different ways and the one you want to use will be different depending on what kind of shape you want to animate).
First, duplicate your drawing (Layer > Duplicate Layer). The bottom layer in the layer list (your original drawing) will be your first frame and you won't need to touch it again. Instead, click on the second layer the second layer. Use one of the select tools to select the pixels you want to animate (either by dragging over the pixels to form a box or border, or by clicking on the pixels - this depends on what tool you're using). You should now have a dashed line around the area you have selected. Next go to Edit > Cut (or press Ctrl + X). This will cut the section you have selected from your drawing. Now you want to add it back to your image by going to Edit > Paste (Ctrl + V). If it pastes as a floating section just go to Layer > New Layer. You'll now have your drawing in three seperate layers. The bottom layer should still be your original drawing, the second will be the part of your image you want to stay in one place and the layer above it will be the part you want to move. You can now move the top layer independantly of the middle one. When you have moved the animated part of your image into the correct position you will need to merge it back down onto the layer underneath by going to Layer > Merge Down. This will combine the two layers back together to give you your second frame.
What you can do now is duplicate the layer again (Layer > Duplicate Layer) and repeat the process to make more movements.
And that's about it! If you need any further help please just let me know.
It's a bit tricky without seeing the icon or your layers to say what the problem is. Flashing would suggest that perhaps you're using layers with some transparency and have your disposal set wrong. If all of your layers are complete and show the entire icon then make sure the disposal is set to replace. If, however, you have a complete base layer and only have the heart on layers above then you need to set it to combine. If the disposal is correct then check all of your layers for any inconsistancies such as stray pixels or even perhaps a blank layer as these could cause the flashing effect.
If you still cannot find the problem please upload the icon and I'll take a look at it.