You need the article because of the restrictive clause. There are two ways to do this:
He has a heart disease which gives him problems breathing when walking up the stairs.
He has heart disease, which gives him problems breathing when walking up the stairs.
The second is the more natural, unless the specific disorder is mentioned.
"Heart disease: Any disorder that affects the heart. Sometimes the term "heart disease" is used narrowly and incorrectly as a synonym for
coronary artery disease.
Heart disease is synonymous with cardiac disease but not with
cardiovascular disease which is any disease of the heart or blood
vessels. Among the many types of heart disease, see, for example:
Angina;
Arrhythmia;
Congenital heart disease;
Coronary artery disease (
CAD);
Dilated cardiomyopathy;
Heart attack (
myocardial infarction);
Heart failure;
Hypertrophic cardiomyopathy;
Mitral regurgitation;
Mitral valve prolapse; and
Pulmonary stenosis."