it would get very upset on cold washes and would possibly have to wait until the water cooled down before it could get on with the wash (and possibly damaging anything in the wash that really needed to be washed in cool water)
You could also end up heating loads more water than is actually needed due to wasted hot water being left in the pipes on hot fill systems.
You "could" place a splitter on the two outlets, to enable cold or hot filling ... BUT ... with both taps open, you would risk cold mains supply back-feeding the hot. You could use a shower NRV in the hot supply to get over this, before the splitter.
If you have a tank cold feed (not rising mains) which is unlikely, you would not get the problem as the head of water is identical.
Dishwasher .. I would just connect the hot. Won't do any harm. I hope it will fill, tho ... Some have an inlet pressure switch. Tank hot will not have enough pressure for this, unless you are on a mains pressure system throughout.