1. It uses a tiny sample size.
2. It assumes American psych undergrads are representative of the entire human race.
3. It uses stupid and incredibly subjective tests, then combines that with cherry picking:
"Thus, in Study 1 we presented participants with a series of jokes and asked them to rate the humor of each one. We then compared their ratings with those provided by a panel of experts, namely, professional comedians who make their living by recognizing what is funny and reporting it to their audiences. By comparing each participant's ratings with those of our expert panel, we could roughly assess participants' ability to spot humor ... we wanted to discover whether those who did poorly on our measure would recognize the low quality of their performance. Would they recognize it or would they be unaware?"
In other words, if you like the same humor as professors and their hand-picked "joke experts" then you will be assessed as "competent". If you don't, then you will be assessed as "incompetent".
Of course, we can already guess what happened next - their hand picked experts didn't agree on which of their hand picked jokes were funny. No problem. Rather than realize this is evidence their study design is maybe not reliable they just tossed the outliers:
"Although the ratings provided by the eight comedians were moderately reliable (a = .72), an analysis of interrater correlations found that one (and only one) comedian's ratings failed to correlate positively with the others (mean r = -.09). We thus excluded this comedian's ratings in our calculation of the humor value of each joke"
The fact that this actually made it into their study at all, that peer reviewers didn't immediately reject it, and that the Dunning-Krueger effect became famous, is a great example of why people don't or shouldn't take the social sciences seriously.