Does the warm water kill bacteria?
Nope. Unless by warm you mean boiling.
To mindlessly massacre bugs clinging on to your cutlery, you should rely on a good quality anti-microbial soap. Not temperature (below 100 celcius).
If I asked the question slightly differently, then perhaps, the answer would be self-evident. So let me pose it again. "Why should we use warm water to rinse dishes?"
Soap or detergent is a surfactant. A bunch of these surfactant molecules aggregate and surround grime and dirt particles, in interesting structures called micelles. The process is well-understood.
Where warm water comes in, is after all the action between soap and grime has transpired. These micelles I was talking about still reside on the dishes, and we need to rinse them off. Like sugar or salt or most other things, these micelles are also more soluble in warm, rather than cold, water. Hence the "warmth" only helps rinse efficiently.
Is this important?
You bet. Because if you can't rinse off dirt from the plates then it remains on the unclean dry plate, right?
PS: By the way, you could substitute "dishes" in the title with "human body" or any other wild-card.