Is the emptiness of persons an emptiness of essence or just substance? Is the emptiness of persons an emptiness of their essence, which (I think) would be a defining characteristic, like how "fire" is always "hot", or just substance?

Do any Buddhist texts say no?
In the Pali suttas, 'emptiness' ('sunnata') refers to emptiness of 'persons' or 'self' rather than emptiness of defining characteristics, which is why the Pali suttas state the five aggregates are empty, or the sense spheres are empty, or the elements (dhatu) are empty. 

'Consciousness' does lose its defining characteristic of 'cognition'; the earth element does not lose its defining characteristic of solidity; the fire element does not lose its defining characteristic of heat; the Nibbana element does not lose its defining characteristic of peacefulness. 


  Insofar as it is empty of a self or of anything pertaining to a self: Thus it is said, Ananda, that the world is empty. And what is empty of
  a self or of anything pertaining to a self? 
  
  The eye is empty of a self or of anything pertaining to a self. Forms... Eye-consciousness... Eye-contact is empty of a self or of
  anything pertaining to a self.
  
  The ear is empty...
  
  The nose is empty...
  
  The tongue is empty...
  
  The body is empty...
  
  The intellect is empty of a self or of anything pertaining to a self. Ideas... Intellect-consciousness... Intellect-contact is empty of a
  self or of anything pertaining to a self. Thus it is said that the
  world is empty.
  
  SN 35.85





  And why, bhikkhus, do you call it form? ‘It is deformed,’ bhikkhus, therefore it is called form. Deformed by what? Deformed by cold,
  deformed by heat, deformed by hunger, deformed by thirst, deformed by
  contact with flies, mosquitoes, wind, sun, and serpents. ‘It is
  deformed,’ bhikkhus, therefore it is called form.
  
  And why, bhikkhus, do you call it feeling? ‘It feels,’ bhikkhus, therefore it is called feeling. And what does it feel? It feels
  pleasure, it feels pain, it feels neither-pain-nor-pleasure. ‘It
  feels,’ bhikkhus, therefore it is called feeling.
  
  And why, bhikkhus, do you call it perception? ‘It perceives,’ bhikkhus, therefore it is called perception. And what does it
  perceive? It perceives blue, it perceives yellow, it perceives red, it
  perceives white. ‘It perceives,’ bhikkhus, therefore it is called
  perception.
  
  And why, bhikkhus, do you call them formations? ‘They construct the conditioned,’ bhikkhus, therefore they are called formations. And what
  is the conditioned that they construct? They construct conditioned
  form as form; they construct conditioned feeling as feeling; they
  construct conditioned perception as perception; they construct
  conditioned formations as formations; they construct conditioned
  consciousness as consciousness. ‘They construct the conditioned,’
  bhikkhus, therefore they are called formations.
  
  And why, bhikkhus, do you call it consciousness? ‘It cognizes, ’ bhikkhus, therefore it is called consciousness. And what does it
  cognize? It cognizes sour, it cognizes bitter, it cognizes pungent, it
  cognizes sweet, it cognizes sharp, it cognizes mild, it cognizes
  salty, it cognizes bland. ‘It cognizes,’ bhikkhus, therefore it is
  called consciousness.
  
  Therefore, bhikkhus, any kind of form whatsoever … Any kind of feeling whatsoever … Any kind of perception whatsoever … Any kind of
  formations whatsoever … Any kind of consciousness
  whatsoever, whether past, future, or present, internal or external,
  gross or subtle, inferior or superior, far or near, all 
  should be seen as it really is with correct wisdom thus: ‘This is not
  mine, this I am not, this is not my self.’
  
  SN 22.79





  And what is the earth property? The earth property can be either
  internal or external. What is the internal earth property? Anything
  internal, within oneself, that's hard, solid, &amp; sustained [by
  nutriment]: head hairs, body hairs, nails, teeth, skin, flesh,
  tendons, bones, bone marrow, kidneys, heart, liver, membranes, spleen,
  lungs, large intestines, small intestines, contents of the stomach,
  feces, or anything else internal, within oneself, that's hard, solid,
  and sustained: This is called the internal earth property. Now both
  the internal earth property &amp; the external earth property are simply
  earth property. And that should be seen as it actually is present with
  right discernment: 'This is not mine, this is not me, this is not my
  self.'
  
  MN 62