Do I need heavy hiking boots for Sinharaja Forest.

You absolutely do not need heavy stiff highly expensive mountaineering boots. In fact they are quite detrimental in the deep jungle humidity as they never dry out. A highly comfortable sturdy pair of standard trail running shoes with a very aggressive deep grip is absolutely perfect. The primary trails are largely flat though occasionally muddy. More importantly you absolutely must wear knee high specialized leech socks pulled tightly over your trousers regardless of what shoes you choose as the damp jungle floor is completely swarming with thousands of tiny active leeches.