generally no, they did not cause more damage to the city than whatever threat they were coming into action against.

in WW2 explosive shells from batteries of AA had fuses -- timed, altitude, proximity. The idea was that flak shells did not need to directly hit the aircraft (very difficult), but just explode in proximity and damage control surfaces and other vulnerable systems on airplanes with flak fragments.

these fuses failed on occasion and unexploded ordinance did fall into populated areas and sometimes exploded on impact or sometime later. generally the damage these munitions did was minimal and vastly preferred over leaving urban centers without a direct response and deterrent to bombing missions.

There are many examples of casualties caused by falling anti-aircraft debris. During both Iraq wars, Baghdad's boisterous AA network reportedly filled the hospitals as thousands ran outside to see/hear the coalition bombs fall many of them being killed or injured from above.

During the UFO scare in LA (Battle of LA), when the military thought they were under attack from the Japanese, there were many injuries from falling AA debris.

Another example comes from World War I when the Germans were bombing Britain (The Zeppelin Raids) with airships and aircraft. The casualties from falling AA debris were reported to have caused as many casualties as the bombings themselves.

Machine guns and Flak batteries both cause damage when they come falling back down to Earth. I could continue naming examples but it is a relatively common and accepted phenomenon and a part of war.