Would forgetting to take my duloxetine for two days make me weirdly moody?

Yes: Duloxotine like any antidepressant can have withdrawal symtoms such as fatigue, tachycardia, exacerbation of depression , moodiness, etc.

AD withdrawal: Stopping any antidepressant abruptly can cause anxiety, jitteriness, irritability, insomnia, etc. And can start as early as 1 day after stopping the medicine depending on which one it is. Some (paxil) are more likely to cause it quickly and more severely than others (prozac).