Are you sure. From my father and grandfather who have had to put down wild animals in the Indian jungle, I have heard the otherway round. Monkeys, dears and birds give characteristic warning calls when they see a tiger on the hunt.
Why would birds go silent when they see a tiger? They're not in danger. The only possible danger would be to their eggs/chicks in a nest and my experience there is they make noise when they perceive any possible intruder near their nests.
Jim Corbett was a Brit in India who had the job of hunting down a number of man-eating tigers, lived to tell about it, and wrote several books about it (recommended). He talks extensively about listening to the jungle as a tiger detection method - the monkeys freak out and call to warn each other about the predator, birds freak out and then disappear, etc. He paid attention to both unusual silences and unusual noises. There were still several occasions where he almost became tiger chow.