McDonald’s Corp. said it plans to appeal after losing some European Union trademark rights to the Big Mac name following a challenge by smaller Irish rival Supermac’s.

The EU’s Intellectual Property Office in a Jan. 11 decision said McDonald’s failed to prove “genuine use” of the trademark across the EU over a continuous five-year period. The Irish fast-food chain filed its challenge in early 2017, the same year that the Big Mac celebrated its 50th anniversary.

The EUIPO, based in...