No, burning and fired are not interchangeable. Fired is not used as an adjective in english meaning on fire. Usually when you say 'fired' in english, it means you were let go from a job. Sometimes you say fired in relation to something being cooked, like clay is fired to make pottery, or a wood-fired grill you cook on. Fired can also mean to use a gun, "I fired the rifle", or referring to a fuel something runs on ("a coal-fired engine"). But it isn't used to mean something on fire or burning.

just adding to Lisa's excellent answer - you can say that a house is/was ON fire. eg "Last night I saw a house on fire at the end of the street." In this example, the meaning is the same as "Last night I saw a burning house at the end of the street."