The teenager, who starred in the CBBC comedy show So Awkward, was found unresponsive at his family home, it has been reported.
Challis, who played the role of Boycie, said his “heart aches” for his former co-star and wife Lucy, who loved their son so much.
He wrote on Twitter: “The saddest news of all today.
“The death of Archie Lyndhurst, just starting out on his chosen career in acting.
“My heart aches for Nick and Lucy.They could not have loved him more.”
Archie followed his father into the acting profession, appearing in Jack Whitehall’s Bad Education series and voicing a character in the video game Final Fantasy XIV: Heavensward, before getting his breakthrough CBBC role.
The children’s channel confirmed the news on Twitter and said the actor died after a short illness.
Cheryl Taylor, head of content for BBC Children’s, said: “All of us at BBC Children’s are devastated to hear the news of Archie’s passing.
“He was such a talented young actor and meant so much to all of those involved in So Awkward as well as to the CBBC audience.
“He will be greatly missed by us all and our deepest condolences go out to his family and friends at this time.”
Channel X North, the independent production company that makes So Awkward, said in a statement: “We are deeply shocked and saddened to hear the news about our friend Archie.
“He was an incredibly talented performer and his contribution to So Awkward, on and off screen, will not be forgotten.
“As well as hilarious, he was a generous, kind-hearted young man who we had the honour to work with on the show for seven years.
“Our thoughts are with his family and friends at this time.”
It is understood that Archie’s death is being treated as unexplained but not suspicious. Met Police have been contacted for comment.
West London Coroners Court confirmed they are looking into the circumstances surrounding the Actor’s death.
A spokeswoman said: “I can confirm the investigation is ongoing and no date has been fixed for the opening of an inquest.”
Nicholas Lyndhurst’s representatives have been contacted for comment.