Here are a few examples of data you might easily encounter that you can pretty much conclude on sight won't have normal distributions, even conditionally: Counts, 0/1 data, continuous proportions, times, Likert-scale data, monetary amounts (like incomes, or claim sizes).
But what I'd love to know if I can look at a set of data BEFORE running OLS and determine if the errors would be non-normal.
You can bet they're not exactly normal in any case.
It's not really the right question to ask though (essentially all our models are not exact descriptions, so it's not actually the exercise here).
In large samples normality may be much less important than the other considerations (the assumption of constant variance, or of independence can be quite important and their impact doesn't decrease with increasing sample size)
For example, would this only occur when the data themselves are non-normally distributed (on one or more variables)?
This is the distinction between conditional and marginal distribution. If you just look (say via a Q-Q plot) at the distribution of your response variable (DV) it doesn't tell you whether your residuals (as proxies for your errors) might be reasonably normal; it depends on how the X's are arranged. You can have normal looking residuals but the DV looks distinctly non-normal -- or vice-versa.
Arthur Charpentier gives some discussion of this issue for both linear models and GLMs here: Simple distributions for mixtures.