Do I need a permit to build a fence?

Not a building permit under the model IRC if it's 7 feet high or less (R105.2). Zoning can still require a fence permit and cap the height, and Birmingham, AL requires one for every fence.

The 2021 IRC's list of work exempt from permit includes "fences not over 7 feet (2134 mm) high" (R105.2, Building item 2). The same section says an exemption doesn't authorize work that violates the code "or any other laws or ordinances of this jurisdiction."

Height limits, setbacks and fence permits come from other local rules, such as a zoning ordinance. That's how a fence can be exempt from a building permit and still need a zoning permit.

Birmingham, AL example: the city's zoning ordinance requires a fence permit, with a fee, before any fence or wall goes up. On residential lots (D, E-1 and R zones), a front-yard fence can be 4 feet high, or 5 feet counting posts and decorative tops, and it can't be opaque. Side and rear fences can be 8 feet, or 9 feet with posts and features, where the neighbor's home sits at least 5 feet from the lot line. Otherwise the city can require a see-through fence such as chain link.

Birmingham also allows chain link only in side and rear yards, bans electric fences, and requires the finished side to face the neighbor and the street. Height is measured from the ground next to the fence to the top of the fence, posts and features. A fence on a retaining wall, terrace or patio counts that structure's height, and a fence in a regulated floodplain needs a separate floodplain permit.

Before you dig, also check any HOA or deed restrictions, utility easements and where your property line actually is.