Why is cigarette smoking dangerous around oxygen tanks?

No, Oxygen is very flammable. Oxygen it self is not flammable in any way; however, it is an oxidizer which means if a fire is started it can very rapidly accelerate the speed at which the fire burns as well as creating a much hotter fire. Think in terms of oxy/acetylene cutting torches, the oxygen it self will not ignite without the acetylene and the acetylene burns at a temp that is too low to cut through metal without the oxidizing agent Oxygen.