When you write up the methods to say you used a particular image, you are providing evidence that you had some copy of it and used it in the way you state. If the copyright holder is highly litigious or offended by your research, this is evidence that you did something that (they may argue) violated copyright. When you rely on copyrighted images, you are also reducing what you can show to colleagues and reducing others' ability to build on you work.
It probably depends on what country you're in and how/why the images were created and distributed by their creator. For instance, if you buy a memory game illustrated with famous cartoon characters, then use that game in your study, you are probably much safer than if you create a comparable game from pictures of those famous cartoon characters. I highly doubt you will get in any trouble unless you're using the images in a defamatory way (e.g. if the memory task induces implicit threats, or if you're measuring which copyrighted cartoon characters are most hated, and that's your main result).
Jim's answer might especially apply if the images were created by a fellow researcher and you are trying to expand on that person's work. There, you are likely to gain permission and gain professional contacts.
Ideally, in experiments going forward, you can avoid any uncertainty by using public domain images or images you made yourself. Then you can publish the images in an appendix, which may even make them standard figures for other researchers to use (and cite) in similar studies. A friend studying elementary school learners described using certain picture books in her research, but to publish her dissertation using key illustrations/page layouts, she had to write for permission and/or find comparable open source images to convey the key points.
If the particular images are extremely important, then you should justify that in the paper, and it may help you if you ever need a fair use defense. It might also make it easier to seek permission from the copyright holders to explain that the quality of their work makes it uniquely valuable for your research. E.g., you're working gaining children's trust across a language barrier and you found that using familiar characters encourages responsiveness. Or if you're looking at people's perceptions of media and you want to do experiments on reactions to real photojournalism, it would make sense to use existing pictures.
I'm not a lawyer, but there may be additional complexities depending on how you use the image. If you have to manipulate the images for your memory tasks (e.g., you edit out a detail and see if people notice), this might be more legally complex with a copyrighted image, since you are making a derivative image. And even how you present the memory task may matter: on paper with original purchased image (e.g. print from a museum store) vs. on paper with duplicated image vs. on a computer (in the U.S., might invoke Digital Millennium Copyright Act).