I think one can argue that “turn on” and “turn off” are familiar phrases, used of radios, TVs, computers, and most other electronic gadgets. Or, for that matter, before radios came on the scene, there was the phrase, “Turn off the lights.”
So, “Do not turn off your computer” seems correct to me.
Growing up a Gospel music fan, the Carter Family's classic Turn Your Radio On, performed here by Mark Lowry and the Gaithers, perhaps should have been titled as "Turn On Your Radio".