What dare of Dill's does Jem finally accept in Chapter 14 of To Kill a Mockingbird?
In Chapter 14 of Harper Lee's To Kill a Mockingbird, there is not the overt dare of Chapter 1, but the reader may interpret one passage as containing a dare of sorts from Dill to Jem. That is, when Dill has run away from Meridian, Mississippi, and caught the train to Maycomb in order to be with Scout and Jem in a home where he is happy, the children discover him hiding under a bed. Jem remarks to Dill that his family must not know that he is in Maycomb:
'Think they're still searchin' all the picture shows in Meridian.' Dill grinned.
'You oughta let your mother know where you are,' said Jem. 'You oughta let her know you're here....'
Dill's eyes flickered at Jem, and Jem looked at the floor. Then he rose and broke the remaining code of our childhood. He went out of the room and down the hall. 'Atticus,' his voice was distant, 'can you come here a minute, sir?'
Beneath its sweat-streaked dirt Dill's face went white. I felt sick. Atticus was in the doorway.
Dill dares Jem not to break "the code of [their] childhood" not to tell on each other to adults. When Jem calls his father, Dill and Scout are appalled that he would do such a thing. Of course, Jem is maturing and realizes that Dill's mother is probably extremely worried about Dill, but Dill and Scout do not yet understand. This action of Jem represents the motif of the novel as bildungsroman, or "novel of maturation."
Although you say Chapter 14 in your question, I think that you are really talking about something that happens in Chapter 1.
In that chapter, Dill Harris dares Jem to go inside the Radleys' gate. He bets Jem that Jem will not do it. Jem does not do it for a few days. Dill keeps making fun of him when he will not go inside the gate. He tells Jem, in a variety of ways, that he is scared.
Finally, Jem decides to accept the dare. He goes through the gate and actually runs up and touches the Radelys' house.