To work in the local tax bureau, do I need to take the civil service exam or the public institution recruitment exam?

Civil Service Examination.

The Local Taxation Bureau is an administrative unit, not a public institution. You must apply for the civil service examination when applying.

Local tax bureaus, referred to as local taxes, are administrative units in nature. Local tax bureaus implement a vertical leadership system of the central government, provinces, cities, and counties. When the tax system reform was implemented in 1994, the original tax bureaus at the provincial level were divided into national tax bureaus and local tax bureaus (some local tax bureaus and finance bureaus work together, called finance and taxation bureaus). Provincial local tax bureaus are under the dual leadership of the provincial people's government and the State Administration of Taxation, and the local tax bureau system below the provincial level is vertically led by provincial local tax authorities.