I'm struggling with severe anemia and am constantly tired. Is it bad to push myself to get a 40 min run in daily?

See doctor. You need to find out the cause of the anemia. Some anemia may be due to bleeding, malignancy, genetics etc... When the cause is found a treatment plan may be formulated by your doctor.
Need balance. First you need to find out the daignossi, implement a treatment plan including diet cahnges and then come in with the exercise part. As your anemia improves you can gradually increase your exercise levels and improve conditionig- not knowing your hemoglobin levels or your baseline fitness level it is hard to comment on specific exercise regimens.