I think that you misunderstand what public recourse funds means. It means that you aren't able to ask the UK government for any money in case you become unemployed, homeless, etc.
Yes, you will have to pay council tax to rent any property in the UK. I did when I was on a similar visa to yours.
I understand that no recourse to public funds means that you are not able to claim those benefits from the government, however I have seen a couple of unconfirmed claims where people have said that no recourse to public funds = no council tax. I know that council tax is used to fund a variety of services but it seems unfair to make you pay a tax which goes towards a majority services you can't utilize (ie public funded housing).
The funding of council tax varies from borough to borough. In general it is used to fund services such as schools, garbage collection, social services, police and fire services, street lighting, etc.
Unless you are a student, then you are exempt from paying council tax. Otherwise, all other UK visa holders must pay council tax.