Looks like the burner wire goes to the Soil Sensing Pressure Switch.
I could not find this switch in the parts so am not sure if your unit even has one.
At minimum you need a new wiring harness (Item 12 in Section 3) and a control board (Item 11 in Section 2).
At a combined price of around $200.00 I would probably get a new dishwasher.
If you have some electrical/electronic experience you could jiggery pokery it.
It is important that you figure out why this wire overheated.
This will take some circuit tracing to figure out what they are doing with this line if in fact you do not have a pressure switch. KUDP01.pdf (981.3 KB)