If this field should show "yes" only when the other three fields are all set to "yes", you need to use AND rather than OR in your calculation:
If ( PostCheck_Pub on Admin="No" AND PostCheck_Rename Campaign="No" AND PostCheck_Remove DraftSample = "Yes" ; "No" ; "Yes" )
Parts that are different from your posted version are in red.
Another way would be to use Boolean expressions. 1 for yes, 0 for no. Set your 4th field as the sum of the first 3 fields. Then you would be able to tell 1) if there is a yes and 2) how many yes's there are. I have been using FileMaker for 15+ years and I just started using Boolean 6 months ago. I wish I had started using them long time ago.
Did not work right away but you gave me enough to understand why. Changed it to:
If ( PostCheck_Pub on Admin="Yes" and PostCheck_Rename Campaign="Yes" and PostCheck_Remove DraftSample = "Yes" ; "Yes" ; "No" )
Now it works like a charm. Did not realize I should be evaluating for yes instead. Thank for your help.