Bottom line is that adding a drop shadow is using a transparency.
If you use Edit>Transparency Blend Space and check if it's CMYK or RGB.
If the image beneath is changing colour, it's because the standard drop shadow effect is set to Multiply (you can edit this in the Window>Effects Panel and double click the Drop Shadow to edit it's appearance)
The drop shadow colour is usually black at 75%, set to multiply (from memory.. I'm probably wrong on that).
Hopefully that gives you enough to go on.
If the image is changing drastically in colour - that's another problem.