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Why do the French say “j’ai chaud”?

For sensations, French reaches for avoir rather than être: j’ai chaud, j’ai froid, j’ai faim, j’ai soif, j’ai peur. You “have” heat instead of “being” hot. The distinction earns its keep, because je suis chaud has drifted: in casual speech it usually means “I’m up for it” or “I’m fired up,” and depending on tone it can be heard as sexual.

Keep the two apart: j’ai chaud (I’m overheating) versus je suis chaud pour un ciné (“I’m game for a film”). Objects work the other way round — le café est chaud is simply hot coffee, and il fait chaud covers the weather. When the heat finally wins, the phrase you want is je n’en peux plus: “I can’t take any more.”