I would have no hesitation in using "burning the midnight oil" even if all I was doing was chatting to a friend late into the night, and I don't see anything wrong with using it to refer to time spent playing video games. Certainly, if I saw light coming from under John's door in the early hours of the morning, then saying that John is "burning the midnight oil" is almost an exact fit, except that the light is coming from a screen rather than an oil lamp.
In the U.S. it's generally used in a context where it's not about entertainment. You wouldn't say you're burning the midnight oil if you are out dancing all night. It is usually used in the context of work or study, and usually it implies the person is staying up late to accomplish a specific task. There is no task to accomplish playing video games for enjoyment.
The origin is interesting and serves as a mnemonic device.
Before electric lights, oil lamps and candles were the main source of artificial lighting. To "burn the midnight oil" meant that someone used a oil lamp through the night. By extension it meant that someone put in extraordinary effort to complete a task.
Nowadays it is a metaphor and interesting that an oil lamp metaphor is being used to describe computer games (which probably don't require any supplemental lighting).