There are multiple possible scenarios. It can either go well or poorly, the main determining factor being the hardware the network relies on. Miners generally only exist for profit. Once every Bitcoin is mined, the miners will no longer have motivation for spending their time and money.
There have been several propositions when it comes to the future of Bitcoin once every unit is mined. The most reasonable proposition was to charge a small fee for each transaction, for the purpose of distributing payment to miners, depending on the computational power they provide the network. It seems reasonable, but there are a couple of important issues that need to be addressed before this was to begin. The Bitcoin concept is solely created for the purpose of offering a decentralized method of payment to a broader range of people. With the instalment of small fees, there would have to be a centralized system that would distribute them. Also, it is important to note miners also invest in Bitcoin hoping for it to develop and become more valuable.
Once every Bitcoin is mined, the metaphor which compares Bitcoin to gold would be even more accurate. Since there is a finite amount of Bitcoin the price will dramatically increase with demand. The key moment will be the period when all the Bitcoins are mined, and this moment will impact the future. Bitcoin could go under permanently or skyrocket and become the most influential method of payment in the world. I believe that Bitcoin is going to survive this critical moment but there are still other issues to be addressed. The network must be upgraded according to the number of users and the computational power that is available.