No, not directly. But when the owners of those properties collect rents, part of the money goes to pay property taxes, which pays for a great deal of the services the city provides, like schools, road, mass transit, etc. If you stop maintaining those services, the city starts to deteriorate and it has the potential to turn into a death spiral--reduced services results in reduced desirability results in reduced population results in reduced tax collections results in further reduced services.
To be clear, I am talking about major downtown areas, like Manhattan in NYC, the Financial District in SF. This isn't about your local High Street. I get that property taxes are different in the UK, but there your landlords pay a pretty whopping rental profits tax (up to 45%), so the concept is similar; if the buildings in the City in London remain vacant there will be a sizable income loss to whichever taxing authority collects that tax. That loss results in reduced services, which will lead to reduced desirability which results in further vacancies and further service cuts.
I don't have a solution myself; I suspect the cities will be looking for handouts from their states and countries, but that is just sending good money after bad. And while I assumed that conservatives and libertarians would take the "Drop Dead" attitude to hearing about the impending demise of SF and NYC, I'm kinda surprised that it's also coming from the liberals and lefties around here.