I know that you're not allowed to eat or wash on Yom Kippur. But suppose a child (not obligated to fast) is going to eat bread - should he wash his hands before eating it?
The Shulchan Aruch (OC 613:1) stresses that only bathing/washing for pleasure is prohibited on Yom Kippur. Elsewhere (Kesef Mishneh, Laws of Prayer 7:8) he quotes the Ran (R. Nissim) who states that any washing (on any fast) that is for a mitzvah one may wash as he typically would. In the case of washing for bread, many Rishonim are of the opinion that washing up to the wrist is obligatory (cf. Biur Halachah §161, end). Therefore, any person eating bread on Yom Kippur should wash as per the usual requirements of "netilat yadayim" - washing up to the wrists. This is the opinion of R. Ovadiah Yosef (Chazon Ovadiah, Yamim Noraim pg. 308 and Arba Ta'aniot pg. 292), R. Nosson Gestetner (Lehorot Nattan vol. 2 §42 no. 3), R. Shmuel Wosner (Shevet HaLevi vol. 8 §139; interesting to note, RSW doesn't even find it necessary to elaborate on the "obvious" answer), R. Shlomo Z. Aurbauch (Madanei Shlomo, Moadim pg. 91), R. Eliezer Y. Waldenberg (Tzitz Eliezer vol. 18 §17).
A minor's age makes no difference in the halachah. The above is applicable to any case where one is halachically allowed to eat (exempt due to sickness, age etc.).