As is true in most cases of comic relief, the existence of a comic character or subplot in a story fraught with dramatic tension helps to establish a pacing for the story's mood.  If too much tension and suspense is built upon the audience, then the later scenes lose their dramatic impact.  However, if the dramatic moments are lightened by comedic elements, then the audience is put at ease.  Then, when the next moment of drama occurs, the mood shift is more dramatic and the scene has more emphasis.

Mistress Overdone, Pompey, and the Constable help to provide this break.  They appear in each act to present contrast to the dramatic characters.  Not only this, but the subplot provides some reinforcement for the themes of morality and justice.  Each character is stuck in a situation based upon the main plot.  Mistress Overdone and Pompey are victims of the new morality laws, and the Constable an unwitting accomplice to the enforcement of these laws.  If this play is to portray the hypocrisy of society and government, than these comic characters provide an undercurrent for that. 

• The jury, passing on the prisoner's life,
  May in the sworn twelve have a thief or two
  Guiltier than him they try. (Act II, scene i)