Whether "burning calories" is used for nutrition or for "burning fat," both terms are common in everyday speech. The rate of "burning" increases during fasting. I have linked one study published in Nature.
Let's go over some excerpts that will answer your questions.
* Metabolism increases during fasting.
> Thus diverse, pronounced metabolite increases result from greatly activated catabolism and anabolism stimulated by fasting.
* The hormone t3 produced by the thyroid is used to control metabolism. In this study they instead looked at markers for actual metabolism.
> We performed non-targeted, accurate semiquantitative metabolomic analysis of human whole blood, plasma, and red blood cells during 34–58 hr fasting of four volunteers. During this period, 44 of ~130 metabolites increased 1.5~60-fold. Consistently fourteen were previously reported. However, we identified another 30 elevated metabolites, implicating hitherto unrecognized metabolic mechanisms induced by fasting.
* The study also mentioned prolonged fasting. Which is earlier stated longer then 24 hours.
> Quantification of blood metabolites from 4 volunteers during prolonged fasting. (a) Experimental procedures employed to study metabolomic changes during human fasting for 58 hr.