Can soap kill cockroaches and/or ants? If so, how?
I'm not completely confident that the aforementioned post is a duplicate, because the question explicitly asks "Why do cockroaches burst from soapy water?" While this question explicitly asks "Can soap kill cockroaches and/or ants?"

The short answer to your question is yes.

According to this website:


  The soapy water kills [cockroaches] by forming a thin film over the
  roach's breathing pores that stays in place due to surface tension,
  causing the roach to suffocate.


Ants are killed in a similar manner (see this archived Reddit post).

For a more scientific read, check out this paper: Behavioral Responses and Mortality in German Cockroaches (Blattodea: Blattellidae) After Exposure to Dishwashing Liquid.