The most important point is that jobs vary significantly by discipline. If you are in the field of Cognitive Science, or in a related field, you are in a better position to know what a job ad in that area is actually looking for. Unfortunately, the job ad linked in the question is written in a way to make it hard for those outside the area to know what it wants - it does not directly say whether they are looking for research or teaching. There is no way to tell without asking what the teaching/research ratio would be.
In general, there are two common uses of non-tenure-track "Assistant Professor" positions in the U.S.:
Postdoctoral positions. These are not usually considered "harmful" to your career. In many fields they are a standard part of the academic job progression.
Teaching-oriented positions for which the candidate has a PhD. These may be full-time positions (as in the linked ad) or part-time.
The second type of position could be helpful, or harmful, depending on what sort of tenure-track position you are looking for. You have already narrowed down the type of institution where you would like to have a tenure-track position, and started honing your CV to be a perfect fit for that type of school, right?
If your type of school is an elite research school, then another postdoc seems more likely to be helpful than a teaching position. This is the type of school where leaving the standard progression is most likely to be harmful to tenure-track chances. If you are a likely candidate for this type of position, you probably know it already.
If you are looking at non-elite public universities, which have more of a balance between research and teaching, a single non-tenure-track teaching assistant professorship is not a mark of certain doom for your tenure track hopes. You can use the job to hone your teaching, move your research forward a little, and you can use the time to apply for tenure track jobs.
If you are looking at teaching-first institutions, or at community colleges, then you need to make sure you have excellent teaching credentials. You might be able to use a teaching-oriented term position as a way to do that.
Unfortunately, because of the excess of candidates relative to the number of tenure-track positions in many (most?) fields, and because the number of tenure-track jobs is not increasing, many academics work multiple term positions in a row. This is especially common in the humanities, and it can be "harmful" for tenure track hopes, unfortunately. I was fortunate to find a tenure-track position, so I can't speak too much to how to handle the situation of multiple term teaching positions, but you can find a lot of discussion about it on the web.