I’m temporarily in China but normally live in another country; can I apply for an Irish passport here?

No, you must apply in the country you are normally resident in.

You should only apply to the Embassy of Ireland in Beijing if you reside in mainland China or Taiwan, or in the case of a lost/stolen passport while visiting here.

If resident in Shanghai or the Provinces of Jiangxi, Jiangsu, Zhejiang and Anhui you should apply to the Consulate General of Ireland in Shanghai and to the Consulate General of Ireland in Hong Kong if living in Hong Kong or Macau.