Rizwan and Irfan both suffered hamstring injuries in the third match in Rawalpindi, where a second-string New Zealand side won by seven wickets to level the five-match series 1-1
Top order batsman Rahul had been put in charge after India decided to rest several frontline players, including regular captain Rohit Sharma and stalwart Virat Kohli, at the end of the Indian Premier League (IPL) last month.
Left-arm wristspinner Kuldeep Yadav will also miss the five-match series after sustaining a hand injury while batting in the nets ahead of Thursday's series opener in New Delhi, the Indian cricket board said in a statement.
All-rounder Hardik Pandya, who led Gujarat Titans to a memorable IPL victory in their debut season, will be Pant's deputy.
The home side have not named any replacements for the injured two.