Looking at the BYU British corpus of written English and also an American English corpus, I found basically 0 instances of "sit/sitting/sits on a computer".
From the feedback I've found here and my own intuitions as a native English speaker, plus my experience teaching ESL, I feel as though "sit on a computer" is safely wrong. It might be akin to "make a photo" another case of absolutely incorrect English that's made like an invasive species among non-native speakers' English.
Much better alternatives are:
1) I've been sitting at the computer all day.
2) If you keep sitting in front of the computer, you'll hurt your eyes.
Also, a previous commenter wrongly suggested that it's like chatting "on Facebook". It's not. We often use "on" to indicate communication through a certain medium, like talking "on skype, on the phone" and "meeting on Tinder". It's different. It's not the same as just being on a computer, when you're just using it.