To clarify what i mean by 'existing normals', in blender you can visualize the normals of faces,vertices etc as little arrows shooting out from them, these exist even when you have never even imported a normal map much less applied it to the object .Do they get replaced, added , multiplied, something else entirely ? I realize this could be a very stupid question if so i apologize.
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http://wiki.polycount.com/wiki/Normal_Map_Technical_Details#Tangent_Basis
So to answer your question: the normals will be altered!
This can be anything from adding, subtracting, replacing, etc.
So practically you could even input an object space normal map for vertex deformed meshes if you do the correct math in the shader.
The common case is to add an offset to them (thus tangent space normal maps)
http://wiki.polycount.com/wiki/Normal_Map_Technical_Details#Tangent-Space_vs._Object-Space