What will happen if I connect hot water to a dish water that is not designed to take in hot water?
Will it break? Or will I just not be able to use the cold programs?
The Siemens SK26E220EU manual says:
Which appears to imply that it is fine so long as the hot water is no more than 60C at input. It includes a water softener, so it may well have some parts in the water path before it heats the water hotter that would be damaged by hotter water. If your hot water is electrically heated, it makes little difference to your overall operation cost - if you have an alternate, less expensive source of hot water, you'll benefit, so long as it's not too hot.
For other models, check the manual - or just hook it up to cold if you are electrically heating your water anyway.
The Bosch SKS62E12EU offers an intensive wash at 70C or about 150F and a 'Maximum temperature for water intake 60°C'. So your answer is hooking to hot water is fine, and will save energy if your central heater is gas fired. Obviously, it can't chill the water, so you'll have only hot wash available.