Can clindamycin cause heartburn and, if so, is this a serious side-effect?

All antibiotics: all antibiotics can cause heartburn,some more that others,specially if it was taking on empty stomach. it is not a serious side effect,not enough to stop the medicine. check with your physician

Possibly: it can cause several GI side effects but they are not usually serious or worrisome

Yes: Take the minimum dose for the minimum amount of time possible. Try taking it with food, sweet potato. Consume a probiotic for a month. Get sunshine (preferably) or take vitamin D3 10,000 IU/day.