Burns will take over the role in Championship and 50-over cricket, with Dernbach appointed to lead the side in the NatWest T20 Blast.
"It has been a great honour to have served the club as captain," said Batty, 40. "The timing is perfect for Rory and Jade to take over and I couldn't think of two better men to be leading the Club forward. I'm very excited for the future of the club."
Burns, 27, came through Surrey's youth ranks and was appointed as vice-captain in 2015. In that time, he has topped 1,000 County Championship runs in three consecutive seasons. He has also agreed a new four-year contract which will keep him at the club until 2021.
"Firstly I would like to congratulate Batts on the way he has captained the club over the last few years," said Burns. "He has given his absolute all to the captaincy, leaving the club and playing group in very good shape.
"I'd also like to thank Batts for giving me the opportunity to be his vice-captain, encouraging me to not only develop as a player but also to develop as a leader.
Dernbach, who led the T20 team on two occasions last season, becomes Surrey's first specialist 20-over captain.
"Jade has always been a leader without a title in our dressing room and out on the field," said Alec Stewart, Surrey's director of cricket. "Appointing him as the Club's first-ever T20 captain is a natural progression in his career.
"As a white-ball specialist, he knows T20 cricket inside-out and his vast experience will give him confidence in this new role. As an Academy graduate and a Surrey man through and through he understands the importance and privilege of leading out a Surrey side."
"Rory has been earmarked as a potential captain for a length of time," Stewart added. "I'm very confident now is the right moment for him to takeover. He will grow into the role while understanding the extra responsibility and workload that comes with leadership. It is a massive honour and I know how proud Rory is to take over."