Sweeping is a two step process.
- Import the paper wallet's key, necessarily compromising its security in the process.
- Immediately send the balance of the paper wallet to a new, uncompromised address.
The second step, sending the bitcoin to a new address, requires a fee which is taken from the balance of the paper wallet. If the balance of the wallet is lower than the required fee for the network to clear the transaction, then it is not possible to sweep the transaction because it will never get confirmed.
The minimum amount you can sweep is one satoshi more than the transaction fee you (or your wallet software) decides to pay for the transaction.
If you have a paper wallet that is too small to sweep, you can still import the private key into a wallet without sweeping. This is more risky because anybody who has or gets access to the exposed key can spend the bitcoin until it has been transferred to a new address. It can be transferred to a new address by combining it with other bitcoin from the wallet to which it was imported as part of a larger balance transaction.