Yes, in some cases
-- depending on how carefully the coffee is reheated, and how sensitive the person is to tannins.
The rule-of-thumb test: if it tastes (unusually) bitter, it's not necessarily good for you.
Image: Bad Coffee at Mulholland Drive by Ricardo Infante, on Flickr; CC-BY-NC 2.0
It's common knowledge that actively heating brewed coffee causes a bitter taste. This is most commonly observed on a coffee brewer warming plate after some time (note that these plates are specifically designed to keep coffee warm and not spoil it). When reheating cold coffee, it should be quite easy to heat a bit too much. This would naturally cause a bitter, unpleasant taste. This is probably the origin of the belief.
Direct heat—e.g., via a warming plate—should not be applied to brewed coffee, at least not for more than few minutes; continued heating will make the coffee bitter. This issue is primarily a concern for autodrip coffee makers; choose a model that dispenses into an insulated carafe over one that uses a warming plate.
I've heard it said that applying heat to coffee causes tannins to form, but can't find a good source. This site confirms that a higher temperature causes more tannins to form. However, the focus is on brewing causing tannin extraction. This should apply to heating afterwards, too: some coffee bean dust (or other small particles) are bound to end up in the coffee. Coffee brewers don't exactly have microfilters.
It's recommended to avoid large concentrations of tannins, as they have many undesirable effects including liver damage in sensitive individuals. So drinking reheated coffee certainly can be dangerous.