Is it a perfect predictor of success? If someone knows you have a low IQ they might not even give you a chance to try.

In the real world this reasoning applies to things like studies on gender differences and performance in different disciplines. Let's imagine there really is a solid scientific case that women are, say, 20% less likely to be good at programming. (Just making up a number.) If this became common knowledge, companies just wouldn't bother hiring women for programming positions. As a result the women who are as good at programming as men don't get a chance.

Rinse and repeat for evidence of racial differences in IQ, etc. If it were in fact true that blacks had on average 5-10% lower IQs than whites (making up another number) and if this were common knowledge, nearly all blacks would face intense discrimination including those with IQs the same as or higher than average whites.

These examples might sound like typical "social justice warrior" fodder. Let me throw out a totally different kind of example to escape that framing.

There's a stereotype out there that male programmers and other male science/engineering types are on average more socially awkward and less socially intelligent than the general population. Now let's say there emerges actual solid scientific evidence for this, and let's say that evidence becomes common knowledge. Would this cause, say, women to just reflexively avoid dating men in technical professions? Would it cause men in technical professions to be passed over for management roles on the perception that they just "don't have the social skills?"

A case can be made that biological differences in ability should be somewhat verboten in public discourse because human beings are terrible at statistics and will lazily extrapolate averages to entire groups. Scientific evidence for a statistical bias in ability that applies to any identifiable sub-group of humans can easily lead to blanket discrimination against that entire sub-group regardless of the strength or mutability of the effect or the long term reliability of the science.

Edit:

The long term reliability issue may actually be a stronger argument than the statistical innumeracy of the average person.

Once something like this becomes "common knowledge" it is very hard to educate people otherwise even if the science is revised. Take fat for example. I still know people who eat margarine because "butter is bad" in spite of the fact that we've known for over a decade that fats aren't necessarily so bad for you and that trans fats (present in margarine) are much worse. Unless you have certain metabolic or circulatory issues butter is much better for you than margarine, but try telling your grandparents.

Let's say some systematic difference in ability becomes common knowledge and then the science is found to be faulty or -- as is often the case with such things -- reality turns out to be a lot more complex than we thought. How many generations would it take to de-educate the population and how many lives would be ruined in the meantime?

We don't generally publish the latest research on biological weapons synthesis or implosion geometries for compact nuclear weapons that use very little fissile material. Studies on race and gender and how they relate to IQ and ability aren't quite as dangerous as these but they should probably be discussed in obscure journals in coded language and should never ever be reported in the popular science press. Society just isn't mature enough to deal with this stuff.

Edit #2:

I'm not arguing it should be illegal to discuss such things or anything like that, just that several good arguments can be made that it's irresponsible to popularize it.