Can we visit to North and East parts of Sri Lanka?

Yes, you can visit. If you are looking to get a cultural experience both area are fine. The cultural experience in these parts are different from the South of Sri Lanka.
But if you are thinking of bird and wildlife watching, the only place I recommend is Mannar. 
This area is a paradise for birders, and I recommend to visit between November to end of March. Thousands of various species of migrant birds can be seen during this period. This area was highly affected by a three decade long civil war and birding was not done during that long period.  
Vankalei Sanctuary in Mannar is a very important wetland site. This amazing birding site was declared a sanctuary a few years ago. You will be able to see “Mannar specials”: Crab Plover, Greater Flamingo, Eurasian Wigeon, Pied Avocet, Common Teal, Bar-tailed Godwits and Black - tailed Godwits, Indian Courser, Black Drongo, Grey Francolin, Long tailed Shrike, Painted Snipe, Golden backed Woodpecker.