Who is the voice of Jesus?

That was my first thought, too: Walter Pidgeon.

It doesn't sound anything like Waltee Pigeon. Pidgeon was a competent actor; whoever voiced Jesus in this film has the distinction of being, in cast full of heavy-handed hams declaiming every stilted line as though they were selling tractors to Soviet farmers, the stiffest ham in the lot, a 19th century orator clearly more interested in the stentorian tones of his voice than in sounding like a real, living human being.

http://www.tcm.com/tcmdb/title/68656/The-Big-Fisherman/notes.html

However, an December 8, 1958 item in Hollywood Reporter's "Rambling Reporter" column stated that the voice belonged to Rev. Donald Curtis.