Suppose we are inferring whether $A$ causes $B$, while holding $N = [N_0, N_1, \ldots, N_n]$ constant and we find $N_i$ correlates well with $A$. There are four reasons to exclude $N_i$:

1. We may have discovered a causal confound $N_i \rightarrow A$ (read: $N_i$ causes $A$) or $A \rightarrow N_i$. If there is a plausible causal relationship (either direction), then $N_i$ should *not* be controlled for, because it is a confounder ($N_i \rightarrow A$ and $N_i \rightarrow B$), a mediator ($A \rightarrow N_i \rightarrow B$).

2. Besides frustrating causal inference, the resulting coefficients will be unstable; doing the experiment again might have very different coefficients for $A$ and $N_i$, meaning the result and effect-size may not be reproducible.

3. Occam's Razor suggests a more parsimonious theory (that with $N_i$ removed) should be pursued first. If $A$ and $N_i$ are collinear, then a theory with just $A$ will be more parsimonious and almost as good as a theory with $A$ and $N_i$.

4. Having that additional coefficient (free parameter) also increases the chance of overfitting, since there are more degrees of freedom, which worsens AIC, BIC, and out-of-sample generalization.

So, why does [Wikipedia](https://www.wikiwand.com/en/Multicollinearity) claim multicollinearity among predictors is not a problem (multiple times in that article) and not a reason to exclude variables, stating

> High collinearity indicates that it is exceptionally important to include all collinear variables, as excluding any will cause worse coefficient estimates, strong confounding, and downward-biased estimates of standard errors.

I agree that blindly removing them _post hoc_ is unjustified, but it should warn the researcher to re-check their _a priori_ reasoning for choosing those variables in the first place, assessing possibility of a mediating and confounding effect and whether the additional parameter is "worth it" in its predictive value.