As the personal owner of (5) forges ranging from gas, propane to coke I will say just leave him alone.It's simply a question of safety, if he is showing proper use, the flames will be very small on coke forges, propane forges are no different than a BBQ grill except the hot box zone and head are made to focus the flame and heat to a very small zone.
I would be more concerned about the hot metal being placed on dry lawn than the forge itself.
Are the next door folks complaining about the banging on the ANVIL or is the person erecting rust color artwork they disapprove of?
I would say more of a zoning issue unless you are going to start coming down on BBQ grills, unless they hooked it up to the main gas line.
JMO - Not a code thing, as most forges are portable in nature.
TOM