Fees won't naturally increase. If I can mine profitably at the minimum amount of transactions enforced by the protocol I'll raise the difficulty as I bring on more miners and squeeze out people relying on transactions to profit. Eventually someone will come along who does mining as a loss leader (a bank?) and will offer transaction fee-less transactions to customers which will squeeze out the last of the profitable miners. Competition will only push the mining profit lower and lower.

Alternatively, a really busy Bitcoin network will be limited by the maximum block size and people will pay for priority inclusion in a block. However, this limitation is totally artificial and doesn't scale with the increasing amounts of bandwidth and disk space available. Miners have an incentive to "suck up" to Bitcoin users - without them their mining has no point. If the Bitcoin users demand bigger block sizes so they don't have to pay transaction fees I think miners will have to take the partial hit from bigger blocks rather than the total hit of lack of blockchain use.