I'd like to play some emulated NES games in grayscale. Is there a video mode/effect I can apply that displays the games in grayscale instead of color?
We normally don't do software recommendations. Because of that, I will explain how to set your Nvidia display drivers to greyscale.
Right-click on your desktop and select the Nvidia Control panel. In the left menu, select the "Adjust desktop color settings" option (beneath Display). then in the main window, under option "Apply the following enhancements", set "Digital Vibrance" to 0%. If you afterwards want to bring back colors, set it back to 50%.
Note that this particular color change is an after effect: It doesn't affect any screenshots you take using the snipping tool or the print screen button, and I'm not sure if it affects video.
This is roughly what your option should look like after this. Note that as I said, it doesn't affect screenshots.
AMD probably has a similar method, but I don't know how to enable it on that platform.
Alternative solution for Windows 10 machines: Open the settings app, navigate to Ease of Access, open Color Filters, then turn on the color filter option and optionally select the Grayscale option. The most recent versions of Windows 10 also allow you to enable a shortcut for this.