Kumar was India's best bowler in the drawn first test at Kolkata's Eden Gardens which ended on Monday and will sit out the remaining two tests in the series.
The 27-year-old was named man of the match after taking eight wickets, including figures of 4-8 in the second innings. In a video posted on social media, Kumar said he was getting married on Thursday.
The second test begins in Nagpur on Friday.
"Kumar will not take any further part in the ongoing test series against Sri Lanka and the all-India senior selection committee has named Vijay Shankar as his replacement," the BCCI said in a statement late on Monday.
"Dhawan is available for selection for the third test."
Left-hander Dhawan made a sparkling 94 in the second innings and his absence will make way for Murali Vijay at the top of the order.
Shankar, who bats in the middle order and bowls medium pace, has yet to play for India.
(Reporting by Sudipto Ganguly; editing by Peter Rutherford)
