Should we include some control variables in regressions when we test a moderation effect of a variable?
Without additional information, it is hard to answer your question but I would say that a moderation model is like any econometric model (and in fact, it is an econometric model).
Therefore, I would tend to agree with @Student, adding control variables is a good idea but it must represent your theoretical considerations (is there an omitted variable bias, a confounding variable, etc ?). So you should only add relevant variables to your model in order to increase its precision.
Just adding random/useless variables will unnecessarily increase the complexity of your model; and as my econometrics professor used to say: "why use a bazooka to kill a fly when you can use a fly swatter ?"
Edit: I (scholar) googled "moderation effect economics" and I randomly chose a paper published in a good journal, you can see that they also use relevant control variables (p. 512).