Though the mission specifically tells you to eliminate the targets, you can tranq them and extract them instead. Doing so will unlock the trophy/ achievement "Depth". Refer to the videos below for more details:
If you can't play the videos above, you can follow the walkthrough below:
Head towards the prison camp and take out the first guard you encounter along the way. You have to take out the soldier guarding the prison camp gate entrance next. A jeep carrying two soldiers will arrive shortly. Hide behind the ledge and wait for them to disembark. They should notice the sleeping soldier immediately so knock them out as well.
Follow the road leading to the main road. There are some tranquilizer rounds inside the guard house ahead so grab them to get your ammo to full then carefully make your way to the armory south of the warehouses. Make sure to knock out the soldier on the watchtower as well. Upon reaching the armory, knock out the guy on the watchtower. By the time you reach the armory, the target should be already on the move. You'll know him immediately since he has two escorts. Tag him and pick the lock of the armory. Pick up Chico's Tape #6 in the corner of the room.
Head towards the West Refugee camp while staying out of sight from the soldiers. The target will walk along the grassy path beside the road. This is where you have to tranq him and the soldier accompanying him. First, tranq the guard walking around the refugee camp below to avoid getting spotted.
Go prone in the bushes and wait until the target starts walking towards you. When in effective range, put them both to sleep, then quickly hide the body of the other soldier in the grass. Another keep will arrive shortly and a guard will get off from it.
Don't make any hasty moves yet. Look around and make sure there are no other soldiers around. Put him to sleep as well and hide his body. Next, unlock the gate leading to the landing zone (spot where you first started in the Ground Zeroes misssion) . Call in the chopper and load Palitz there to complete your first objective.
The second target is patrolling found within the walls of the admin building. After extracting Palitz, the base will go on alert status since they spotted your chopper. Continue carefully towards the northwest entrance of the admin building. Tag/ knock out any enemy soldier you encounter so you'll have an easier time extracting the other target. Once you've reached the northwest entrance of the admin building, just reload the checkpoint and the target will start with his original position on the wall. Put the soldiers on the ground to sleep then wait for the target to climb down the ladder. Put him to sleep and his escorts who will climb down after him. Carry the second target to your preferred landing zone and call in the chopper to extract him. Mission complete.