The two-part programme, which was due to air on Monday, sees eight members of the public try to enter the UK from Europe by evading border checks.
A spokesperson said: "In light of the tragic events we have postponed the transmission of the series.”
Smuggled had been described by the broadcaster as an "unprecedented national security experiment”.
A press preview screening for the two-part show was scheduled for 23 October; however, it was cancelled earlier in the week for unrelated reasons.
Police were called to the lorry in Essex yesterday morning where the bodies were discovered, with all 39 people pronounced dead at the scene.
Officers have said the vehicle had travelled from Zeebrugge, Belgium, into Purfleet, Essex.
When Smuggled was announced in August, Channel 4 said it wanted to explore Britain's border security "as the UK prepares to ‘take back control’ of its borders and end freedom of movement for EU citizens".
Meanwhile, Head of Factual Entertainment Alf Lawrie said at the time: "This dramatic series has real purpose at its heart. Extreme, authentic and utterly timely, it is a standout experiment that we hope will inform the national conversation.”
Promotional material for the show, made by production company ScreenDog, said: "From boats to lorries, ferries to car boots - all of the contributors attempt routes that have been used by those seeking to enter the country clandestinely or by people traffickers and drug dealers."
With reporting by PA.