Which patients should not have an angiogram?

Non-surgical. Patients who are unwilling to have surgery even if needed should not have an angiogram because there are risks with this invasive procedure that would outweigh the zero benefits in this setting of no plan to act on the obtained information.
Depends. You should not have an invasive examination if the diagonsis can be made with a non-invasive examination, or if you are not will to persue treatment regardless of the findings. Angiograms are very helpful diagnositic tools and treatment depending on the abnormality and doctors skills can be done at the same time.