The balloon is not a hot air balloon which require a burner as a heat source. None was ever shown.
In the castle - notably in the scene with the gorillas - a switch-box and electrical conduit are clearly visible on the wall. The story takes place in 1878, long before such electrical equipment would have been installed.
During the balloon preparation scene, one of the native characters in local dress hurries by the camera revealing blue jogging shorts with white pipping down the side showing from under his costume. Very popular style for the 1960's but definitely not available in during the Victorian Era.
In several places the shadow of the boom crane that is used to suspend the "balloon" is visible as it supposedly floats in the air.