For that voting power to exist, you’d need to be able to exercise it and/or have someone in the appropriate jurisdiction with power who would back up your claim to do so.

Realistically, that isn’t happening anytime in the foreseeable future.

I still don't get how this is a logical path. If BP wanted to hurt Rosneft wouldn't keeping the shares but announcing that they saw them as worthless except as a tool to influence russia be more impactful? Or any other path that would indicate their lack of trust in the company but still retain a way to influence it if that door opened again?

Even if BP considers the shares and the voting power completely and utterly worthless the other owners of Rosneft apparently don't and giving back the shares gives the other shareholders larger ownership of the company for free.

BP, per the article, has not decided what they will do exactly - “[…] without saying how it plans to extricate itself.”

They just don’t want to be involved anymore.

Realistically if they keep their shares, they’ll get hounded by everyone to do said influencing or whatever, which would be really irritating for them I imagine. But maybe they will put them in a drawer somewhere in the basement and tell everyone to screw off. Who knows. They say they don’t know right now.