Can you tell me if "of" is necessary here, please? :
I dont' remember how much we have to pay. Can you remind me (of?) the amount of our bill?
When followed by a noun (or noun phrase) it is needed. However with different phrases following it's not.
'Can you remind me how much it is?'
In the example you gave, you can also leave out anything following at all:
'I forgot how much we have to pay, can you remind me?'
(not a teacher, just a language lover)