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I’m trying to understand how the cash taken out on cash-out refinance is taxed (or not) when the property is sold.

It's not. The cash taken out is irrelevant for tax purposes.

For example, I invested $50K into a rental property over a decade ago.

Irrelevant. What is important is the price paid for the rental property, not the cash invested.

I refinanced a few years and I took out $75K cash.

That doesn't matter.

Say if I sell it now and make a gain of $100K, I understand that I have to pay capital gains on that.

The gain is the difference between the sale price and your basis in the property. Since this is a rental, your basis is the original purchase price plus purchases costs, add in any improvements made over the years, and subtract off the depreciation that you took, or should have taken, on the property. Cash invested, cash out refinances, and cash proceeds from the sale don't enter into the picture.

How does the $75K cash that I had taken out play into it and how (if it is) is it taxed?

It doesn't enter the picture.

The upshot of all of this is that it is quite possible that your cash out refinances can leave you without enough cash after the sale to pay the taxes due on the sale. That can be a very uncomfortable position to be in.

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