Broken banknotes (euro or old D-Mark) can only be exchanged at the Bundesbank. It doesn't matter if it's torn, torn through or half burned. As long as more than half of the bill can be presented, an exchange is not a problem. If there are fewer remains, proof is required that the missing portions of the note in question were actually destroyed. The damage must not have happened intentionally.
Coins can also be exchanged, for example if their value is no longer recognizable due to many years of use.