Adobe Indesign CS6. Take a look on this image:
How to remove shadow under blue shape?
I can’t clear effects on this object:
The drop shadow appears on both objects because the anchored object is part of the main text frame object.
Instead of applying the effect to the text frame object, apply it to the fill. Since the fill is contained in the text frame object, and not the anchored object, the shadow will appear only behind the text frame. (Note that this only works if the text frame actually has a fill, but a Paper fill works just as well as a color or gradient.)