Does lymphoma cause trouble breathing?

It can. If lymphoma is involving either the chest, neck, or even abdomen lymph nodes (pushing up on diaphragm) it certainly can. Also if the lymphoma is causing anemia that is another cause. Lymphoma can release cytokines (chemical that spread) that can cause fatigue, fever, and (presumably) shortness of breath.
See below. It can if it affects the lungs or if lymph nodes are compressing something or causing a pleural effusion.