It seems to have originated in the US with Fire Departments:

> These reports show that a dry run in the jargon of the fire service at this period [1880s–1890s] was one that didn’t involve the use of water, as opposed to a wet run that did.

https://www.worldwidewords.org/qa/qa-dry1.htm

Interestingly the one place I have seen "dry run" to actually mean "dry run" is using a air compressor to check to see if a water loop (in a computer) doesn't leak by seeing if there no drop in pressure.