OpenGL is generally faster on Windows now than Direct3D11. I would assume that OpenGL on Linux generally performs at the same level of Windows or better (there is the notion out there that Nvidia spends a good amount of time and effort into making sure it's Linux driver performance is good). Again, this is all broadly speaking, there are some cases (see below) where OpenGL isn't faster than Direct3D11. I can't speak for AMD or Intel as it concerns Dolphin performance, however.
To be fair, we've yet to do a benchmark quantifying any hard data, so we're just eyeballing things at the moment and going by experience. Personally, OpenGL performance has never been an issue on Linux, except when it comes to EFB2RAM. EFB2RAM seems to negatively affect OpenGL performance, and on certain games it's quite noticeable (Metroid Prime, New Super Mario Bros. Wii). Direct3D11 (and 9) don't suffer the same performance loss as OpenGL from what I've heard.
To be fair, we've yet to do a benchmark quantifying any hard data, so we're just eyeballing things at the moment and going by experience. Personally, OpenGL performance has never been an issue on Linux, except when it comes to EFB2RAM. EFB2RAM seems to negatively affect OpenGL performance, and on certain games it's quite noticeable (Metroid Prime, New Super Mario Bros. Wii). Direct3D11 (and 9) don't suffer the same performance loss as OpenGL from what I've heard.