If a zombie dies without player contact, will it drop a "rare drop"?

Solution 1:

Unfortunately, no they will not drop rare drops without player intervention.

According to the wiki, in order for a zombie to drop an iron ingot, it must be slain by a player or tamed wolf:

They can also drop one of the following when killed by a player or tamed wolf:

• Iron Ingot
• Carrot
• Potato (baked potato when killed with fire)

However, the odds are low, and baby zombies don't drop them at all:

Baby zombies cannot drop the three items mentioned above when killed by a player or tamed wolf, but they still drop 0–2 rotten flesh after dying.

This drop has a 2.5% chance of occurring, increasing by 1% per level of looting. Individual items have the following chances of dropping:

• 1⁄120 (about 0.83%)
• 7⁄600 (about 1.17%) with Looting I
• 3⁄200 (1.5%) with Looting II
• 11⁄600 (about 1.83%) with Looting III

Similarly, equipment also has a low chance of being dropped:

Zombies have an 8.5% chance of dropping their naturally-spawned equipment and drop it with a random durability. Each level of Looting increases the chance by 1 percentage point (11.5% with Looting III).