If I have a single unspent output with 1 btc, and I send 0.3 btc to my friend, what happens with my 0.7btc?
It gets sent to my adress right? But why does this sometimes cause problems and paper wallets have a warning on them to always spend the full amount
?
Because, if you are not familliar with the technical details of the software that you use to send the transaction, then the 0.7 BTC might be sent to a different address than you expect.
Some wallet software will generate a brand new address with a brand new private key, and store the information about that address and key without telling you about it. Then it will send the 0.7 BTC to that new address. That address and key are not on your paper.
If you think you are done sending the transaction and you delete the wallet from your computer, then the address and key will be gone. You will have lost the 0.7 BTC permanently.