the lord of the rings - Why is Aragorn shaking while holding the Palantir and talking to Sauron?

He appears to be in pain while holding the Palantir.

Why is this so?

Answer

For reference, this is the scene in question:

In the actor commentary on the scene, from the DVD, the palantír is described as a "direct line to evil"¹.

It's worth comparing Aragorn's experience to Pippin's, from earlier in the film:

Basically as soon as he touches the Stone (or as soon as he's communicating with Sauron), Pippin is clearly in excruciating pain. Clearly there's something deeply uncomfortable about having a psychic phone call with Lucifer.

This is borne out by the text, although bear in mind this is information Peter Jackson could not have (legally admitted to have) used as reference:

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  Using the Stones is a mental workout:

  
  
  

  
  

  The use of the palantíri was a mental strain, especially on men of later days not trained to the task

  
  

  ^{Unfinished Tales Part 4 Chapter III: "The Palantíri" Note 13}

  
  

  
  


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  Using the Stone against Sauron is a particular strain. We learn this in the context of Denenthor's encounters with Sauron:

  
  

  
  

  The breaking strain of Denethor's confrontation of Sauron must be distinguished from the general strain of using the Stone.

  
  

  ^{Unfinished Tales Part 4 Chapter III: "The Palantíri"}

  
  
  

  
  

  This meshes well with the description from the commentary, of the stone as a "direct line to evil".

  
  

Aragorn is at an advantage because, as the Heir of Elendil, he's the lawful owner of the Stones, while Sauron is not:

In the case of Denethor, the Steward was strengthened, even against Sauron himself, by the fact the Stones were far more amenable to legitimate users: most of all to true "Heirs of Elendil" (as Aragorn), but also to one with inherited authority(as Denethor), as compared to Saruman, or Sauron.

^{Unfinished Tales Part 4 Chapter III: "The Palantíri"}

However, he's still a mortal going up against a god-like being. You've got to expect some effort from a confrontation like that.

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¹ Thanks to Valorum for pointing this out