If it was attached to a location that did not produce income would it still give -1 gold? For example if a plot gave 4 gold would the player only get 3 if they had a burned and pillage on a location?

Yes.

Just as a character without a military icon would gain one with rusted sword and +2 strength, a location without a gold production would gain "-1 gold and -1 initiative" when burned and pillage is placed on it.

The "-1" modifiers gained are modifiers in their own right. They do not rely on any other text or characteristic of the attached location to work.