Yes and No.
No: where the member of staff's role is directly funded by the precept e.g. the Clerk, then the council will not suffer a funding cut, therefore government advice is for the entire public sector is not to furlough, and it could be investigated by HMRC.
Yes: when the member of staff is funded primarily by other funds, e.g. lettings fees / sales, then they can be furloughed, bar staff, community centre manager / cleaners, caterers etc.
It could be possible for someone to have two contracts with the council, e.g. a park keeper (a role covered by the precept) to keep working, but have their community centre cleaner role furloughed as the level of cleaning is linked to (and funded by) the number of lettings that occur in the community centre.