Wordorigins.org

A dry run is a rehearsal. The term appears to come from American firefighting jargon, where dry denoted a practice where the hoses were not turned on.

Run, meaning a response to a fire alarm, either real or a drill, dates to the late 19th century. The Portland Morning Oregonian of 11 September 1886 contains a use of wet run in reference to a contest between fire companies:

Open to all; wet run; distance, 200 yards to hydrant; lay 350 feet of hose; [...] ; attach pipe and throw water.