Local governments outside Manila have refused to accept the roughly 6,700 tonnes of rubbish the capital generates every day.
Here, 'refuse' is used as a verb. To refuse means to say 'no' to a request; you deliberately do not do it or say firmly that you will not do it.
The noun is 'refusal'. Example: The refusal of local governments outside Manila to accept the roughly 6,700 tonnes of rubbish the capital generates every day has made the problem worse.
In the wake of the Payatas disaster, there were bold pledges to clean up open dump sites and find safer areas for the city's refuse.
Here, 'refuse' is a noun, but it has nothing to do with saying 'no' to a request. It's a formal word referring to rubbish, waste material and all the things that are not wanted and thrown away.