The internet is a unique thing to civilization, with a practical and an idealistic definition which people interchange at will. Classically the internet is a collection of servers all over the world sharing their information through high speed wireless, phone lines, and satellites. In this sense, it's entirely possible that the internet will outlive us all as this technology is the base for so many other things; mobile phones, universities, research, libraries, media, advertising and much more. However there is another side of the internet which most people know; the free sharing of ideas and information, available to all as a human right.
Servers are built to last, and because so many empires are built upon them, it's likely that they will exist in one way or another throughout the rest of time; a way of keeping networks connected. Many companies or organizations have an internal internet, such as learning portals or back-end things. These are generally known as intranet, and are the basis for even less connected things like deliveries and retail stores. Their own private internet, not connected to the external internet except by monitored passages or gateways.
For this aspect of the internet, it's like saying 'will phone lines last forever?'. They require maintenance, and some places do not have them, but the concept of them will likely last for as long as we do. They are a convenient, easy to install method of transferring data, and much more is based off of phone lines than you might expect. Though satellites may have electrical disturbances such as solar flares, the basest level of technology in phone lines can still be used. This is, experts believe, how the internet will work once the newest technological revolution comes about. This happens all the time in history, such as with telegrams; the submarine lines are still there and usable, but people do not anymore because better options have come about. Even more recently, with the change from using 2G to 3G to 4G on mobile phones; the systems are still available, just disused. The base of the internet will always be around, because information still needs to be shared worldwide in this day and age.
As for the popular aspect of the internet as a free-speech centre, the future is much less certain. Already, organizations are attempting to privatize the internet for their own gain. The most recent example is obviously America and their Net Neutrality debating, where senators of states were lobbied by ISPs such as Comcast to repeal the laws keeping the internet free. Of course, people still have to pay for the internet, but everyone should have to pay a flat price to access the entire internet, while Comcast is trying to get people to pay more to access certain free sites. Net Neutrality may well be the downfall of the internet, and it doesn't look like people have much of a say in the matter.
Internet censorship is likely just the first step in the downfall of the internet as we know it, and many suspect that in future years people will look back on these years as the 'golden years of the internet'. This is already seen in some, less democratic, countries such as North Korea, where they have a country wide private internet/intranet known as Kwangmyong. Here they do not have access to the global internet, and can only use the propaganda filled intranet available to them. For North Korea, the golden age of the internet has already gone, and it appears as a matter of time before organizations are doing the same.
Odds are, we will always have access to 'an internet', but we won't have access to the full, unabridged internet we know today. Fortunately for the savvy user, the dark web is much less regulated, and using a browser such as Tor can get around most of these roadblocks. Or you could spoof your IP to that of a more-free country. But let's face it; you shouldn't have to.