In general, it would not be advisable.
DI water is used for a couple of reasons. First, the missing ions aid in the cleaning process, as they seek to bond with the material on the board, allowing the water to act like a fairly aggressive solvent. Secondly, the deionized water should contain no ionically active contaminants. Pure water is non-conductive, but, the various minerals and contaminants contained in the water is. Depositing these conductive contaminants on the board could lead to the potential for down-stream issues with the boards.
In past lives, I've had customers who ionically tested boards as they were received in house...even washing with mostly depleted rosin beds would produce failures. RF boards, current sensitive circuits, and high-voltage/power boards are all susceptible to circuit failure that can be directly traced back to washing with "ionized" water.
You should consider analyzing why the wash machine is consuming a lot of energy (if it's consuming energy beyond what it's rated at). You could also analyze (or have your DI resin supplier) analyze the utilization pattern of the water. It's possible that your wash/rinse cycles are set longer than is necessary, consuming an unnecessary amount of water; or that the resin-bed system that was installed does not have an adequate amount of capacity to handle your actual needs, and needs to be upgraded.
cheers, ..rob
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