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.)
Shogakukan's Kokugo Dai Jiten Dictionary backs up senshin's answer, listing the following for the term's etymology:
「焼け処(ど)」の意。「やけと」とも
Meaning 焼{や}け処{ど} ("burnt place"). Also read as yaketo
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:
やけどを漢字で「火傷」と表記するのは、意味からの当て字。
やけどの「やけ」は「焼け」、「ど」は「所」「場所」などを意味する「処」で、「焼け処」が語源である。
井原西鶴の浮世草子『好色二代男』に「脇腹を見たまへば、焼所ありありと」とある。