Are there any mitzvot that kids under bar/bat mitzvah are obligated to keep? (not including for "educational purposes")
A child that has not yet reached the age of maturity is not obligated to keep any mitzvah. There is an essay by the Steipler Gaon (perhaps in Kidushin) that entertains the idea that kids are obligated Rabbinically to keep mitzvos and that this is the definition of the mitzva of chinuch. I don't believe this is a generally accepted opinion.
The obvious problem with this approach is that if children have no obligations, then they aren't obligated to listen to the sages either. It's a nice discussion, and if I find it I will post the exact citation.
The Pri Megadim in the Pesicha of Chelek Bais in Ois Gimnel says that Ketanim are obligated in negative (Lo Saase) Mitzvos. Harav Moshe Zatzal in Yoreh Deah Chelek Bais Siman Ches disagrees with the Pri Megadim.