If my score is good and if I'm applying for an MS program in Europe, can it give me any kind of edge? The university I'm applying to requires GRE scores for certain programs but not the one I'm applying for. And if it won't be advantageous, can it hurt my prospects in anyway?
Generally speaking it's okay to add whatever to your CV as long as it does not distract the reader from more important points/experiences.
For a GRE score, I highly doubt it would hurt to put it in, even if it is not required. Not to mention that some may perceive this as a plus.