have the fire是这个状语从句里的谓语和宾语,而burning all the night只是个宾语补足语
“ have +sb./ sth.(宾语)+ 现在分词(宾语补足语)”意为“叫 / 让 / 使某人做某事或让某种情况发生”.宾语 sb./sth .后面用现在分词作宾语补足语,表示宾语与现在分词表示的动作之间为主动关系,且动作正在进行.例如:
Don't have the baby crying!不要让婴儿啼哭!
Don't have the dog barking much,Lilin.李林,别让狗狂吠不停.
You'd better have your car running slowly.你最好把车子开慢点.