The answer ot that is "yes" and "no".
Senior royals who are on "the royal circuit" of official duties are always given an "Official London Residence" on the Royal Estate (ie - in one of the Occupied Royal Palaces). The cost of maintaining these buildings is paid by the Government by way of the Property Services Grant in Aid (in exchange for the Queen's hereditary revenues from the Crown Estate).
All other internal costs (domestic and office staff, catering etc.) used to be paid by the Government from the Civil List, but today the Queen pays that herself from her Duchy of Lancaster income. This extends to the Queen's four children, as well as three of her four paternal first cousins (the Dukes of Glouchsters and Kent, and to Princess Alexandria). Charles does receive an Official Residence, but no Civil List (his household is funded by his Duchy of Cornwall income).
So in summary - when Diana was married to Charles, their apartment was an "Official Residence". The physical structure was paid for by the Government through Property Services, while the household costs were funded by the Duchy of Cornwall. At the divorce, Diana was granted the apartment herself by the "grace and favour" of the Queen. The Government still maintained the building itself, but Diana's household costs were funded by the 600,000 pounds annual household allowance that was part of her divorce settlement. I don't know who paid that allowance (The Queen herself or Charles, but either way it was a private - not taxpayer - expense).