Answer
No, a video card is not required to be installed on the server machine. Some ArcIMS ArcMap Server and ArcGIS Server features require OpenGL 1.3 Library functionalities. These libraries are typically included in the operating systems in which ArcGIS Server is supported. In Microsoft Windows operating systems, these libraries are included with the operating system.
On the client machine, using hardware-based graphics accelerators can improve dynamic 2D and 3D rendering performance; however, all video cards may not work. If hardware accelerators for ArcGIS Globecontrol and ArcReader 3D are being used, make sure the graphics card is compliant with a minimum OpenGL 1.3, and be sure to use the latest available driver.