Why is "他在椅子上坐" wrong?
1) 他坐在椅子上  = 2) 他在椅子上坐着 


  He is sitting on/in a chair. 
  
  or
  
  He sits on /in a chair.


You choose which translation to use depending on which part you want to stress.(movement or status)

3) 他在椅子上坐 (questionable)


  He sits on /in a chair.


If He sits on /in a chair in English sounds right to you, then 他在椅子上坐 is grammatically correct as well.

Here I can give you a rare context where it CAN be correct.


  你都这么大了，行为要变得正常一点，在桌子上吃饭，在厕所拉屎，在床上睡，在椅子上坐，该怎么样就怎么样。


You'd better put your context here so we can discuss. Safely speaking, avoid the third usage unless you are confident that you are correct.