Leather shoes are not allowed to be worn on yom kippur or tisha b'av. What if both the sole and the upper materials are made out of synthetic or non-leather materials but some of the shoe's design is made out of leather on the sides? This is commonly found in running sneakers.
Rabbi Ari Zivotofsky states at Tzarich Iyun: Leather and Fasting on Yom Kippur that any leather attached to the shoes will make them forbidden to wear on Yom Kippur or Tsh'ah B'Av See note six below.
Reprinted from JEWISH ACTION Magazine, Fall 5772/2011 issue
Normative halachah maintains that “shoes” refer to footwear that includes leather, even a small amount. Accordingly, footwear made only of cloth (e.g., sneakers) or any other material may be worn on Yom Kippur. But, as noted, there is an opinion that any “protective” footwear is prohibited. According to the Rambam (Hilchot Shevitat Asor 3:7), innu’i implies that one must feel the ground and sense that he is barefoot. The Bach (OC 614) testifies that several of his teachers would walk completely barefoot on Yom Kippur, and he ruled accordingly. However, the Magen Avraham and Taz write that the generally accepted custom is to permit non-leather shoes. The opinion that prohibits any protective footwear is also cited by the Sha’arei Teshuvah (OC 554: 11) and the Kaf Hachaim (OC 554:72). The Sha’ar HaTziyun quotes the Chatam Sofer that when walking in the street on Yom Kippur one should wear thin shoes so as to feel the ground and sense that he is barefoot. The Mishnah Berurah (614:5) concludes that if possible, one should follow the strict opinion in this regard and not wear protective shoes.
. Shulchan Aruch, OC 554:16 and 614:2 following the Rif, Rosh, and Tur. Rabbi Ovadiah Yosef (Yalkut Yosef, Moadim  p. 90) permits wooden clogs with a leather strap. Although in this regard the accepted halachah is that “shoe” is synonymous with “leather shoe,” that may not be true for all halachot. For example, the SA (OC 4:18) requires one to wash hands after removing one’s shoe, and Rabbi Yitzchak Nissim, a former Israeli chief rabbi (1896-1981; Yein Ha’tov 1:13), discusses whether washing is required also for merely touching one’s shoe and whether the halachah applies to all shoes.
. One must be careful not to mistakenly wear sneakers with leather sides.