How should I handle a paper with incorrect experimental results?
In my opinion, the right thing to do is to publish a paper to clarify what could have been wrong in the previous work. This adds to the knowledge and invalidating previous results/beliefs is part of the science. 

It is a good practice to inform the authors of your findings if you want to make sure as you said: 


  We also don't have any reason to believe that the unreproducible
  portion of their results is incorrect as well


By informing the authors, they might provide you with more information that might invalidate your conclusions.  

Assume good faith and accept that humans do mistakes.