When the entire volume of bitcoin is mined there won't be anything left behind for mining. By such situation, the earning of the miners will be through the transaction fee. This will be decided by the miners, if there isn't any combined decision then the transaction fees rising is unstoppable.
Good point. It makes sense for the network to rise in fees, in order to sustain miners' operations. After all, miners will earn solely from fees on the Bitcoin blockchain once the max supply is reached. By the time fees are ridiculously expensive, it's expected that the Lightning Network would dominate most of Bitcoin's transaction activity. Those willing to use Bitcoin as digital cash for daily payments will be force to use the Lightning Network, while those willing to use it as a store of value will have no choice but to perform transactions on-chain for maximum security. It's the way Bitcoin is heading to, and there's nothing we can do to stop it (unless developers raise the block size to solve the high fee issue).
Nonetheless, there's no need to worry about ever-rising fees on the Bitcoin blockchain. As I've said before, people have the option of using the Lightning Network for cheap, instant payments. If that's not enough, there's a plethora of altcoins to choose from with ample network capacity for everyone. Coins like Bitcoin Cash, Litecoin, and Dogecoin have ridiculously low fees making them practical for day-to-day transactions. What matters is that Bitcoin remains decentralized and censorship-resistant in order to stand the test of time. High fees tells us that Bitcoin is doing well in this regard. As long as it stays decentralized, nothing else matters. Just my thoughts