Does having a second monitor affect gaming performance?

The short answer: Yes

The long answer: Running two monitors simultaneously stress out a GPU as now it has to render a larger resolution. To understand this better, imagine you are using a FHD (1080p) monitor and you decide to add another FHD monitor, now your GPU has two FHD monitors to keep up with. Due to this, your FPS will definitely be worse off than the FPS you were experiencing when your GPU had only one FHD monitor to keep up with.

Depending on the type of GPU you are using, the drop in FPS can range from nominal to significant.