Given that やけど means "burn", it seems likely that the やけ part is 焼け. What about the ど, though?
(The kanji for やけど - i.e. 火傷 - don't help, because those characters are obviously 当て字, and would ordinarily be read かしょう.)
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It looks like やけど derives from 焼け + 処, i.e. "burnt place", and was sometimes also written 焼（け）所. cf. 語源由来辞典: