As a tropical country, it is very fair to say that Sri Lanka does not have any seasonal changes like in Europe. You can watch resident birds and endemic birds throughout the year. We have two different monsoon rains. But it is not a big hindrance to watch birds. From October winter migrant birds start coming to Sri Lanka. These winter migrants can be seen until April. If you are concerned with both resident and migrant birds to Sri Lanka, the best time to visit is November to April.
You can read great some good information about Sri Lanka Weather, climate and geography on worldtravelguide.net.