The tripods were not immune to Earth's diseases. It wasn't neccisarily the birds that killed them off, but more so they germs they held.
***END SPOILER*** [/QB]
The birds had nothing to with that, the only significance of the birds is that they landed on one of the tripods i.e. the army could see that the forcefield wasn't working properly, and so were able to take it down.
"Just When I Thought I Was Out - They Pulled Me Back In....."
