The Shulchan Aruch writes in Hilchos Shabbos (343) that it is forbidden for an adult to "feed" something which is forbidden either by the Torah or M'drabonon, to a child. This concept is not based on the mitzvah of chinuch, but is rather based on a specific prohibition of לא תאכילם, from which the gemara learns out in Yevamos 114a - להזהיר גדולים על הקטנים. The Rashba cited by the Biur Halacha (quoting R. Akiva Eiger) disagrees and holds that if it is only an issur d'rabonon and being done for the sake of the child, not for the sake of the adult, it is permitted.
There is a major machlokes Rishonim addressed by Tosafos 77a, and by the Ran and Tosafos Yeshanim at the beginning of the perek whether the other inuyim (aside from eating and drinking) are forbidden by the Torah, or only m'drabonon. Rashi 74a d.h. shabason, seems to hold that they are only assur m'drabonon because Rashi equates the d'rabonons of Shabbos for which "Shabbason" is used as an esmachta, to these inuyim for which shabbason is also used as an esmachta. Yet, when explaining our gemara which says that one cannot put shoes on children because people will say אינשי עבדו ליה - an adult did it for him, Rashi writes that the Torah forbids an adult from feeding an issur to a child. Therefore, if we allow children to wear shoes, people will assume that an adult violated this prohibition by putting the shoes on the feet of the child. Even though Rashi holds that these inuyim are only M'drabonon, and even though it is being done for the sake of the child, Rashi assumes that there would be a violation of לא תאכילם - להזהיר הגדולים על הקטנים. This seems to be against the Rashba and is a support for the Shulchan Aruch who holds that even for an issur d'rabonon done for the sake of the child, we don't allow an adult to be the direct cause of it.