Summary: Armored AA-carriers are explicitly excluded, and any other tank without a cupola-mounted MG can safely be described as lacking any meaningful AA capability.
Some tanks mounted MGs that were widely used as AA gun (.50 cal, 12.7mm), but those only really threaten low-flying and lightly armoured planes.
The answer is "not", as german WW2 anti-ground tanks such as the Tiger/Panther did not have any.
All German models shared the basic layout of having a primary gun, along with coax and/or chassis mounted MGs.
Gun elevation specs for the tiger can be found at http://www.tanks-encyclopedia.com/ww2/nazi_germany/Panzer-VI_Tiger.php
Upwards elevation was limited to 15 degrees. Transverse speed beyond the first few degrees would be bound to either the chassis, or the turret.
Maybe some tank somewhere scored a lucky hit on a low-flying aircraft, but that would inspire tall tales told at camp fires.
Edit: noticed now you weren't exclusively asking about German tanks. Answer still stands for comparable designs, but some allied tanks featured an MG on top of the turret that could actually point upwards.
Efficacy for these would be comparable to other emplaced machine guns.
Edit2: As for any tanks equipped with MG's mounted on the commander/loader cupola: the AA action summary of 1945 by the US Navy mentions that
Although the .50-caliber machine gun destroyed more enemy planes than any other weapon at Pearl Harbor, it long had been appreciated that the weapon was obsolete because of its short range and light hitting power.
These statements are backed up by figures showing that .50 and .30 cal got very few kills while firing a lot of rounds. (.30 cal guns on US navy ships used 56,950 rounds to bring down a single plane in 1942)
Late-war Russian tanks (eg IS-2) would be equipped with a DShK AA gun, which reportedly was reasonably effective against low-flying aircraft.