In Chinese mainland, Father's Day (the third Sunday in June) is almost unknown. It isn't a public holiday, but expats in China might celebrate it. Perhaps some Western fathers might be honored on the day by their family, friends, or workmates.
This wasn't always the case though. During the WWII years, Father's Day was celebrated on August 8. The Republic of China government wanted to celebrate the soldiers who died and honor fathers. They chose that date because in Chinese August 8 can be shortened to "bā bā" (八八 'eight eight'). This sounds similar to the informal word for father (爸爸, bàba). This tradition dropped off in the Chinese mainland, but was continued in Taiwan under the Republic of China government.
Taiwan's Baba Jie
In Taiwan, Father's Day is not an official holiday, but is now widely observed on August 8. People often take their father out for dinner, give gifts, or at least call them to give them a greeting. (Baba Jie means 八八节 or 爸爸节: 'August 8th Day' or 'Fathers' Day'.)
Macau and Hong Kong
In Macau, Father's Day (Dia do Pai) is the third Sunday of June, and it isn't a public holiday. In Hong Kong, Father's Day is celebrated on the third Sunday of June and is not a public holiday either. Local Hong Kongers have adopted Western traditions, so it is more common for Hong Kong fathers to be greeted, treated, or honored on this day.