If a person had lung cancer, they couldn't feel the tumor when they take deep breaths, could they?

Difficult breathing?: Not really sure if you mean as with difficulty breathing or be able to externally see/ feel it ( which would involve more the skin). In either of these cases, see a doctor. But' just taking deep breaths would not typically allow you to feel a lung tumor.

Usually not: Most lung cancers start in the interior tissue of the lung where there are no pain nerves. If the tumor, or its metastases, involve the outer lining of the lung(where there are pain nerves) , one can then feel pain on breathing. Other medical problems (like pleurisy) can also cause pain on breathing