In addition to the dried out riverbed, a dry run can also be a dry enclosure for poultry to run in. But for practice meaning, Etymonline.com says "Dry run is from 1940s" and there's plenty of US military examples of a practice without ammunition during WWII.
Military: firing practice without ammunition
An American Speech from 1943 defines the term:
'DRY RUN (to practice; a dress rehearsal).' I never heard it used as a verb, or to mean a dress rehearsal. Originally a semi-official term for practice firing without ammunition, it is slang in other senses, such as a mail-call at which one receives no mail.
From a 1932 Coast Artillery Journal:
Later we had a short practice with the Navy; a so-called "dry run," which proved not so dry.
From a 1932 The Military Engineer description of 1918:
The French had designed a trestle for a bridge over the river and one over the canal and delivered them to us in trucks ready framed for erecting. Many dry-run erections were made on dark nights until the structures could be erected in ...
Also some army test flights, including this non-military 1934 test flight in Multiple lens aerial cameras in mapping:
Climb to altitude and if the area is near at hand study ground and map flight lines, while climbing. 5. Reaching altitude, make a trial or "dry" run. This enables the pilot to find his drift and the photographer to find the time between ...
(There are later test flights in in 1938's The principles and practice of surveying and 1941's Photogrammetric engineering and remote sensing.)
Here's a theory with not much to back it up. The banks of dried riverbeds, dry runs, were useful as tactical cover for infantry. This 1914 Infantry Journal says:
In framing problems, take as few liberties with the landscape as possible. It borders on the ridiculous to state in a maneuver problem that a dry run is an unfordable river or that a hill is a marsh.
It's a stretch, but were dry river beds often used for infantry practice maneuvers? Did this lead to dry run becoming a practice without ammunition?
Another theory goes:
As per the etymology for "dry run", I wonder if the roots to its artillery past aren't stronger than what is given on the site... When muzzle-loading cannon is fired, the bore is swabbed out with a wet sponge on a pole so that the embers from the shot that was fired, do not prematurely ignite the gunpowder that is being loaded for the next shot. Of course, if the exercise is just a drill, there would be no need for a wet sponge. A dry one would be used. Hence, "dry run"?
I should add that when a firing range on a military base is ready to start shooting, the Range Commander asks for permission to "go wet". A flag is raised and the shooting starts.
I can't find any evidence for this, but there's definitely a strong military connection.
Firemen's races: practice run without water
However, the fireman theory is also good. The military use is practice firing without ammunition. The firemen's use is a practice run without water.
During firemen's races, each part of putting out a fire became its own event: a dry run: "running out the hose" over a distance and connecting to a hydrant; a wet test: "throwing out water" with given pressure over a distance; a coupling contest: attaching hose to the hydrant; a ladder climbing contest.
The 1894 The New York clipper annual ... containing theatrical, musical and sporting chronologies ... includes three instances of dry run in firemen's races, for example:
dry run, 200yds. to hydrant, attaching, unreeling 350ft. of regulation hose, breaking coupling and putting on pipe, each three full turns, cast 515lb, stripped, pipe 24[?] long, 51/2lb, carried by pipeman, who started with team.
So they run with the hose and attach to a hydrant, but in each, no mention is made of turning on the water. The same publication details other races describing a certain steam pressure and water pressure.
A 1910 The Harvester World details a similar Fire Department Contest, with a similar Dry Run event, with other events: Ladder Climbing Contest, Coupling Contest, and a Wet Test involving "throwing water".
The 1916 Tacoma: its history and its builders; a half century of activity
Eagle Hose Company of Old Tacoma won the "wet test" in 38 1/2. seconds. The "wet test" consisted of running 660 feet, connecting with a hydrant and throwing water. The "dry test" was won by a Vancouver, BC, company in 4 1/2 seconds. A squabble arose in the firemen's contests over the right of Struve, one of the Seattle firemen to participate, it being charged that he was a professional runner, which in fact he was. Some two years before he had run a race in Tacoma with Halstead, and had carried a 50-pound sack of flour on his shoulders as a handicap.
The British Fire Prevention Committee's 1907 Red Books tabulates the results of dry and wet tests. It also talks of a "for position of hydrants and run of hose, see Fig. 7." and several uses of "the hose was run out in [a direct line/the same direction]". So run appears to be the technical term for laying out hose, and perhaps why the US races are generally either "dry run" or "wet test" (in addition ladder climb or coupling contest) - each part of the drill is separated out.