The system of beehive-like tunnels stretch more than half a mile beneath the cemetery and have been excavated over the past two years.
The church leaders have mounted the protest following a court ruling against them over the sale of three Greek Orthodox buildings.
Israeli police chased down and shot and killed the gunmen involved in the shooting at the holy site.
On April 6, a ritual sacrifice took place inside the walls of Jerusalem's Old City for the first time in 2,000 years.
Men charged with "disturbing the peace" at the religious site.
The bloodshed has raised concern of a wider escalation of bloodshed.
Israeli police are investigating suspected Jewish vandalism of an abbey near the Old City.
Mazen Qaq has had a series of threats against him since he put up his reward.