Why doesn't the plate heat up?
The heating plate no longer gets hot and the water doesn’t pump to the grounds.
There are different series of DeLonghi coffee preparing automates. Look on the serial plate, mostly on the bottom. The ESAM family identifier starts either ESAM 3xxx, ESAM 5xxx or ESAM 6xxx.
Here’s a link to the DE' LONGHI ESAM2000 - ESAM3000 - ESAM3100 - ESAM3200 1 SCHEDA TECNICA ST1462/2008 ESAM4000 - ESAM4200 service manual You didn’t state the model number of your coffee maker ;-)
To download the file, when the sentence below the Document preview box changes from This file is downloadable free of charge (5/day): ...processing... to This file is downloadable free of charge (5/day): Download, click on Download, as this is the link. Be patient, it takes a little while to find the file.
The manual has schematic diagrams which may help you to test the plate and also the pump/pump thermal protector to see if they are still OK or not.
It also shows how to get into the diagnostic test mode to test the various functions of the machine.
Hopefully this is of some help.