Compatriot Donncha O'Callaghan will take O'Kelly's place among the replacements for the clash in Rotorua.
Head coach Sir Clive Woodward said: "It is obviously disappointing for Malcolm but this is why we have the strength in depth of the touring squad.
"It enables us to provide cover to ensure that we get the players back to full fitness needed to tackle the tough games ahead."
Meanwhile, Woodward has confirmed Jason Robinson will arrive in New Zealand on June 7, in time to play against Wellington three days later.
The England full-back stayed behind when the Lions left for New Zealand to be with his pregnant wife.
"Jason Robinson is arriving here on June 7, 100%. I spoke to him yesterday. His wife is a lot better. He can play in the Wellington fixture," said Woodward.
And Scottish loose-forward Simon Taylor is likely to make his first start of the tour against a Test-strength New Zealand Maori side, set to feature the likes of Carlos Spencer.
Taylor injured his hamstring as the squad prepared for their send-off game against Argentina, but Woodward said he might feature against Taranaki on Wednesday and is certain to get a run next weekend.
"Simon is coming on well. He pulled a hamstring and will possibly be OK for selection next Wednesday but will definitely start in the Maori fixture."