"Jem opened the box.  Inside, surrounded by wads of damp cotton, was a white, waxy, perfect camellia.  It was a Snow-on-the-Mountain...
Jem picked up the candy box and threw it in the fire.  He picked up the camellia, and when I went off to bed I saw him fingering the white petals."

Imagine having a neighbor whom you believe made it their personal mission to torment you.  That was Ms. Dubose.
I believe according to Atticus, you never mock the dead.  No matter how terrible they were in life, they have it far worse in death.  They greatest gift is life, so why despise those who had that gift taken from them?
And to have them give you something that says they wanted you to know they thought highly of you before they died, shows true character.  But to then rip up that memento is even more disrespectful than mocking the living.
Jem could have just as easily taken the flower and cherished it but instead made the ignorant decision to burn it. I essence, he burned one of the only remaining earthen forms of Mrs. Dubose's spirit.
That shows a lot of what kind of simple minded person Jem is.
But of course youth is always accompanied by ignorance.