My older dd had a blocked tear duct until she was 8 months old. We did warm compresses and massage, but it didn't help. We even took her to a ped. opthamolgist who taught us how to properly massage it (massage right above the little bone on the inside of the bridge of their nose.... the tear duct is actually down further inside the nose... massaging it is supposed to create a suction to release the tiny membrane blocking the duct).
We ended up taking her to a Naturopath who gave us a homeopathic remedy(Silica 30C). We gave her one crushed tablet once a week for 4 weeks and it competely cleared up!!! (You can do every week until it clears up, but it's supposed to take about 4 weeks.) It had NO side effects either!! You can buy the Silica at Whole Foods or other places that sell homeopathic remedies. My ds has blocked tear ducts, too, and I just started the Silica with him yesterday!