Unfortunately, Asia's wildlife (unique or otherwise) has long been in decline due to poaching by traffickers, killing by rural farmers, and habitat destruction.
Sri Lanka has the largest remaining elephant population. The best national parks to see them include Yala, Minneriya, and Udawalawe among others.
Tigers are more elusive and you'd have to be extremely lucky to see one. Populations are scattered thru-out South Asia in India, Nepal, and Bangladesh.
The sad fate of orangutans in Indonesia is known. Another simian population are the snow monkey macaques of hot-spring fame in Yamanouchi, Japan.
The Papua New Guinea highlands for exotic birds.