Oddly, the Oxford English Dictionary doesn't include an explicit definition for the contemporary sense of an elimination round, giving instead a 19th- century sporting usage which suggests a single competition broken into different stages. It isn't clear when that semantic adjustment took place, but the most obvious original source for the current term is horse racing, in which a "heat" was used to describe a run given to a horse purely by way of exercise before a big event - what would now be known more temperately as a warm-up.
But there are other meanings, too, which seem to lend extra poetic force to the current usage. Heat has long had a connection with physical excitement, sometimes sexual (to be "on heat") but by no means always. And a "heat" could also mean a single intense exertion; to do something "at a single heat" was to do it in one go, a meaning which probably derives from foundry work, in which a "heat" was the term given to any one of the multiple heating operations applied to the ore. Each successive heat improved the metal, so it isn't entirely fanciful, perhaps, to see some notion of refining or annealing fire preserved in the athletic sense. The term might even be taken as a good example of a linguistic serendipity, with several overlapping usages blending together to create a verbal alloy that is stronger than any one component.
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