If I have a paper wallet that has .001 btc, could this be worthless in the future because when I sweep the funds the fee is so high?
Paper wallet is a generic term.
You might be talking about a single address (single private/public key pair) or a seed (with billions of addresses)
If you have 12-24 words, it is a seed. So you don't have to worry about and you may transact between your generated addresses freely.
If you have only one address, you will have different inputs/outputs from the same address, if you send funds to another address your paper wallet becomes worthless