There are a few different mechanisms that could explain why people thought repeatedly-seen headlines were less unethical to share.

Fluency: Even though participants were told the information was false, recognizing the headline could have made it feel more intuitively true. This could explain the weaker moral condemnation in the intuitive thinking group in experiment 3.

The fluency-truth association can make the blatantly false information easier to digest. Alternatively, we have weaker reactions to stimuli as they occur repeatedly. This could be happening with the headlines. Both would explain why seeing a headline once led to less of a moral condemnation reduction than seeing the headline 4 times.

It is important to note that the reduction in moral condemnation is not a result of the illusory truth effect. The illusory truth effect does not apply to clearly false information, and the subjects were clearly told the headlines were fake, except for experiment 4. The participants needed to pass a comprehension check that ensured they knew the article was fake, so they could not be basing their judgments on thinking the article was true.