Sri Lanka experiences a warm and tropical climate for much of the year. Even peak heat doesn’t reach the child-unfriendly temperatures of 35°C, and, while it can be humid; it is entirely manageable, especially as most accommodation offers a swimming pool and aircon, so you don’t have to worry about your kid overheating.
As far as diseases go, though, rabies and animal-borne parasites are present in Sri Lanka, so keep children away from stray animals, including cats, dogs and monkeys. Always take precautions against mosquitos!
Don’t forget to get travel insurance for your child as well as for yourself.