When one is in remission, is that person still taking chemotherapy?

Maybe. Some cancers "in remission" are still treated with therapy. Examples include: myeloma (lenalidomide), indolent lymphoma (rituximab). Other studies are looking at "maintenance" therapy in other disease. Examples: lung cancer - cox-2 inhibitors colon cancer - statin drugs aggressive lymphomas - lenalidomide also, adjuvant (post-surgery) chemo is given to patients with no evidence of disease (ned).
Not necessarily.... But they could be. Simply inducing remission may not be enough to stave off disease recurrence. Post-remission therapy is given to reduce risk of recurrence after apparently having eradicated one's disease. It's use reflects the realization that people can harbor cancer cells even in remission. Continued chemo is given to address this. Remission does not equal cure...Hence the two terms.
Maintenance chemo. What you are asking about is maintenance chemotherapy. Continuing chemotherapy after a complete response is dependent upon the cancer.