What's the safest treatment for lymphoma patients who already have heart disease?

That is a tough. Question as there are many type of lymphoma. In a patient with cardiac disease you would try to limit the exposure to anthracyclines -- the most common being doxorubicin.
Important question. It depends on the type of lymphoma, need for treatment, type of treatment and type of heart disease. One of the drugs use in the treatment of lymphoma (doxorubicin or adriamycin) can be problematic in some patients with certain types of heart problems. Most other treatments can be done without difficulty. There is quite a bit of experience with this, as well as guidelines.