Narine, 35, announced his retirement from international cricket in 2023, four years after playing his last match for West Indies against India
Tharanga will miss the next two matches of the five-match series, in Pallekele on Aug 27 and Colombo on Aug 31, after Sri Lanka were adjudged to have bowled three overs short of their target in the specified time.
It's the second time Tharanga has been suspended this year, after missing two matches in June's ICC Champions Trophy for the same offence of his side's slow over rate.
Sri Lanka brought in Dinesh Chandimal and Lahiru Thirimanne to the squad in place of opening batsman Danusha Gunathilaka and the suspended Tharanga.
The hosts lost Thursday's match by three wickets and trail India 2-0 in the series.