It's not the painting at issue, it's the reproduction. Photographs of paintings have copyrights that belong to the photographer. If you have access the the old painting and take a picture of it, you control the copyright of that reproduction.
I think the confusion here stems from the fact that that statement assumes you don't have access to the painting, which is the most common case.
Wikimedia Commons disagrees. I know they're not arbiter of law, but you'd hope they have it right given the scale of how wrong they'd be if they were wrong.
> Photographs of paintings have copyrights that belong to the photographer.
This is highly dependent of the country that you are in. In the case of the US, the courts have ruled that a faithful reproduction (eg photograph) is not original enough to merit a separate copyright (see my like to Bridgeman Art Library v. Corel Corporation above).
Additionally, the post is not saying that you are required to use the exact photo found in the out of copyright book, but rather that you are required to find a photo (any photo) of the same painting in an out of copyright book.
I think the confusion here stems from the fact that that statement assumes you don't have access to the painting, which is the most common case.