In Johnny Tremaine, what is the significance of Lieutenant Stranger's torn up letters?
The answer to this can be found two or three pages into Chapter IX. The significance of these letters is that they tell the Patriots when the British are planning to move against them.
At this point in the book, Johnny is working as something of a spy for the Patriots. He has some contacts who work among the British. In this case, his contact is Lydia, the black woman who does laundry at The Queen. She has found letters that Lt. Stranger has been writing to Lavinia.
In the letters, he has been trying to tell Lavinia why he can't go to a dance with her. One of his attempted letters shows that it is because the British army will be marching out to the north on that day.
So the letters show something about the British plans and that is important to the Patriots.