I want to apply a Chi-Square test, but I'm not sure whether to include or exclude non-responses. For example, I want to see if GENDER (male, female) is correlated with SATISFACTION on some public policies (Satisfied, Dissatisfied, No answer). Should I use a 2x2 table (excluding No answer) or a 3x2 table (including No answer)?
The 'correct' answer will depend on what your interpretation of non-responses are. Your research question does not seem to really include "No answer" as a component. If you can assume that there are no gender differences in what is meant when individuals respond "no answer" then, to directly address your question, you can chose to eliminate that group and just do your 2x2 $\chi^2$. I'd suggest a little more due diligence. Before you do your proposed 2x2 $\chi^2$ (Satisfied/Dissatisfied x Male/Female), why don't you do one that is (ProvidedAnswer/NoAnswer x Male/Female)? If this second $\chi^2$ yields no differences then you at least no there is no effect of gender on the non-response rate. Although this doesn't solve the potential confound of genders implying different things when they didn't respond - it is a step in the right direction.
P.S. If you perform the 3x2, how would you interpret the resulting statistic? It wouldn't unambiguously answer your question as a significant statistic could be arrived at just if there were gender differences in the proportion of "No Answer" responses.