In this image, I want to keep the drop shadow, but not the white background.
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Unless the drop shadow and the white background are on separate layers, then your only real choice is to make them both transparent, and then re-create the drop shadow.
Remember to create the new drop shadow on a separate layer, or you'll have the same problem again if you want to change the color of the background or the drop shadow.
The problem you'll have is that the drop shadow is a mixture of both the color of the shadow (black) and the background, and they can't be easily separated. So if you try to change the background to transparent or say, blue, what you'll be left with is the content surrounded by a grayish drop shadow, further surrounded by a blue background. Yuck.
But creating a new drop shadow is fairly easy. (These are generic instructions as I use Gimp at the moment) Use a selection tool to select the content that you want to cast a shadow, and copy it to the clipboard. Create a new, transparent layer, and paste it into that later. Turn all non-transparent pixels black (PS should have a "paint whole selection" function). Use the blur functions to "fuzz" up the edges of your new shadow, use warp functions to bend the lower corners downward and the upper corners inward, and voila - a new shadow. Now you have to create an Alpha channel for the drop shadow layer, and then create another layer that is the color for the drop shadow.
Again these are generic instructions but I hope they will illustrate the basic concepts.
Hope that helps.
Actually I've found the best way to do this is to select all colour to be removed with the magic wand tool. You don't need to be too pedantic at this step, just make sure you get all of the large areas.
Next, invert the selection (Ctrl + Shift + I on Windows), then from the 'Select' menu, choose "Refine Edge...".
In the Refine edge dialog, I found the "View mode" thumbnail "On Layers" to be the most clear.
Tweak the other settings, making sure you choose the "Output to" to whatever you're after, I went with "New layer with layer mask", and 'Ok'. Voila, pretty much perfect removal.
Also, I found the best way to see my refined selection was to have a solid fill colour layer beneath the image you're removing colour from (I was removing white, so my layer was black).