Yes. In the context of skating, I'd say it increases bias, in fact, since more judges might want to score a certain way so as not to have their scores thrown away (for instance) - tending them towards a popular opinion about someone's skating, say.
I do not think that this is much of a consideration in the minds of the judges. There is no shame in having your score disregarded in the final tally for being the highest or the lowest. (After all,
somebody's score has to be highest and somebody's the lowest.)
Usually when this type of criticism is made of ISU judging, the focus is on "corridor judging" rather than on trimming the mean. The presence of the "corridor" might psychologically push a judge to score according to his guess about how the other judges will score the event. This bolsters "reputation judging," which is unfortunate but perhaps inevitable in the context of figure skating.
By the way, the theory of statistical inference does not give us very much help in discussing the trimmed mean. For instance, all of the formulas that have sigma/sqrt

in them are inapplicable in the most general case. There is, in fact, no completely satisfactory technique for dealing with this problem, and in circumstances where it is really important to get it right, statisticians must rely on "boot-strap" procedures tailored to the particular data.
In the case of symmetrically distributed data it doesn't matter. Any method of trimming or not trimming gives the same result. (This is approximately the case with judges' scores in figure skating). For an unsymmetric distribution, one must know something about the amount of "skewness" in order to make suitable corrections (but if you know all anout
that you basically already know the answer and you don't need statistics at all.)
In figure skating, all is well. The distributions of scores, even when judges cheat a little, is sufficiently "Gaussian" that is doesn't matter statistically. The only situation where it might make a difference is the one that I mentioned above, where one datum is a true humdinger of an outlier due to a typo, not just someone trying subtly to game the system.