I run our tap hot at home simply because the water is heated by the sun, not gas, electricity or anything else. So it pays to save the wear and tear on the dishwasher (Besides, it still works out to save electricity, with water costs factored in).
The results are generally no different, aside from drying, which I'll explain below:
Our machine (DishDrawer) doesn't shorten the cycle times as a result of this, as the temperature of the wash phase is only boosted about 10ºC more than room temperature (about a 20ºF raise).
For the rinses, I run the tap again. The machine will retain the temperature, to a maximum of 50º (122F) - as thats what the water is tempered.
Although, if I've run a very hot cycle, the machine stays hotter. This often results in the Final (Heated) Rinse not being heated... Generally, if the temperature of the machine is within 5ºC (inclusive) of the target, the heater does not engage*
I think only the Normal/Normal Eco, Heavy/Heavy Eco and Fast/Fast Eco cycles shorten things if the temperature is reached sooner than expected.
The Normal cycle should wash for about 35 minutes, but only washes for 26 in every instance I've noted (whether cold inlet or not). I can usually stop the Eco variant of this cycle (Which washes/rinses at 104/40C) from heating at all!
The Heavy Cycles often don't heat the final rinse water, due to the higher temperatures retained from the wash.
Fast Cycles don't shorten much more than a minute or two.
*This is observed with a diagnostic display mode, which will display temperature in the units for your country