I can't remember what happened to the child of Professor Remus Lupin and Tonks after they both died in the battle at the end of Harry Potter and The Deathly Hallows: Part 2. Did Harry take care of their child?
According the Harry Potter wiki, Edward "Teddy" Lupin was raised by his maternal grandmother, Andromeda Tonks, after the death of his parents in The Battle of Hogwarts.
Not much of this is mentioned in the films themselves if I recall, as such a lot of the information will come from the book and/or Pottermore.
So no - Harry, Teddy's godfather, did not raise him.
He was raised by his maternal grandmother Andromeda with help from his godfather. He began attending Hogwarts School of Witchcraft and Wizardry in 2009. During his time at Hogwarts Teddy was Sorted into Hufflepuff house and in his seventh year was appointed Head Boy.
Check Wikia for more details