The title says it all. If it wasn't maintained for hundreds of years, would the Channel Tunnel collapse?
Thanks to SeanC's research, we can read that the "permanent fixings" should have a lifetime of ~120 years. I'm uncertain as to what sort of error bars are on that figure, or whether there's a good safety margin, but it is a good start.
Once the lining starts breaking down, the tunnel will become unsafe. Most of the tunnel is bored through marl, a relatively soft chalky rock, which doesn't bode well for its very long term survival. Long-lasting chalk excavations do exist, but the examples all seem to be relatively shallow implying there's not a lot of overburden compared to the Channel Tunnel, which is obviously underneath the sea as well as a load of rock and crud. The tunnel was designed to cope with a certain amount of cracking and water ingress, but there will obviously be limits, especially over a large timescale.
So I'd guess that a century is probably fine, but a millenium is too long. I'm sure you could handwave something inbetween. One thing to consider is that collapses may only be partial (remember there are 3 bores in total, with links between them) and the rock isn't super strong so some re-excavation may be possible, given suitable equipment and knowhow. Certainly, I wouldn't expect the whole thing to just fold flat suddenly one day.
For the now deleted subquestion: the tunnel is about 50km or 31 miles long. How long it would take you to walk would depend on your fitness, and the condition of the floor after whatever time has elapsed. A day isn't a bad ballpark figure, but I recommend trying a walk of that distance for yourself, just to see!
Zeiss Ikon's answer is of course absolutely correct . Apparently, the pumping systems for the tunnel were built to expect 20 litres per second per kilometre of seepage. The total volume of the three bores is about 5.5 million cubic metres. So it'll take a couple of months to flood. Once submersed, I suspect the 120 year guarantee might be voided.
I do note though that the maximum depth of 40m is pretty tame by mining standards, and you would not need a powerful and modern pumping system to drain it. Primitive steam should be quite enough, if you wanted to do such a thing. Additionally though, the same source that provided the 120 year lifetime figure also says that "internal caulking" should have a 20 year lifetime, and be accessible, presumably for ongoing maintenance and replacement, suggesting that pumping alone is not enough and the tunnel would need active maintenance.
I suspect that a flooded tunnel would be restoreable, especially by anyone with a laissez-faire attitude to health and safety. Though the lining would likely saturate, the metal reinforcements require oxygen to break down and that's going to be in short supply after a while. It might not be very safe once drained, but active shoring and re-excavation as the water levels drop seems achievable. It seems unlikely to be doable by a low-tech pre-industrial society, however.
Collapse of the tunnel isn't what would make it impassible first.
The tunnel itself might well last a century or so, but if there's no electric power for as little as a few weeks the tunnel will be closed by water seepage that can't be pumped out.
This water entry happens in all tunnels that go deep enough under the local ground surface – mountain tunnels in Switzerland, for instance, despite being a thousand, or even near two thousand meters elevation, are sloped so the water that comes in will run out on its own. This is obviously not helpful with an underwater tunnel; the center of the tunnel, more or less, must be the lowest point. Underwater tunnels, then, must solve the seepage issue with pumps – and if the pumps fail (due to lack of power to run them), there's nothing else that can do their job.
I don't have exact figures for the seepage rate in the Chunnel, but you ought to be able to look up its pumping capacity (thanks to @MorrisTheCat, this article quotes 1000 m³/hr), which should be designed with a safety factor of at least three (or a redundancy level at least as good – three pumps, for instance, of which any single one can do the whole job). From that knowledge, you can make a good estimate of how long it would take the tunnel to flood after the last pump quits. Beyond that, it doesn't matter much how long the structure and liner last, the tunnel would have to be pumped out from the ends and/or ventilators just to begin to restore its own ventilation and pumping systems.
Others have suggested that windmills (a la Dutch land reclamation with medieval technology) could either keep the tunnel pumped out, or pump it out after it has been flooded for some prolonged period. Given the above figure of roundly a million liters per hour seepage, even when the tunnel is open and under constant maintenance, and the figures I've been gifted for the amount of water removed from the Beemster when the Dutch were pumping it down (comes to around a billion liters a year for a single windmill), plus the approximately 50x greater head required to pump water out of the lowest part of the tunnel, it would take around 450 windmills (equipped with many kilometers of lossless power transfer hardware, since the actual pumps would have to go down in the tunnel) just to keep up with the seepage the current pumping system is designed to handle. A hundred or so extra could then pump out the tunnel in a matter of some years.
Unless, of course, there's been even a single, small, partial collapse anywhere below water; such a failure would allow effecively (relative to the ability of pre-industrial equipment to deal with it) unlimited water to flow in as water is removed, rendering the tunnel permanently closed.
For a differing point of view, Alan Weisman wrote a book called The World Without Us, speculating on what would happen to our built legacy if humans suddenly disappeared from the Earth. He noted, for example (before hurricane Sandy) that some subway systems, like those in New York, would flood within days without active human intervention (i.e. continuous pumping). He also dealt specifically with the Channel Tunnel, in the very long-term question of whether Britain could be colonised via that route by new terrestrial animals from Europe, in the context of accounting for ongoing sea-level rise.
His contention was that portions of it would likely flood, making it a block to animal passage (both terrestrial animals and fish, although I guess amphibians could make it), but that the structure itself would likely survive for a very long time:
Should some impetuous animal attempt the journey via Chunnel— the English Channel Tunnel, Le Tunnel sous la Manche—after human traffic ceases, it might actually make it. Even without maintenance, the Chunnel wouldn’t quickly flood like many of the world’s subways, because it was dug within a single geologic layer, a bed of chalk marl with minimal filtration.
Whether an animal would actually try is another matter. All three Chunnel tubes—one each for westbound and eastbound trains, and a parallel central corridor to service them—are swaddled in concrete. For 35 miles there would be no food or water—just pitch darkness. Still, it’s not impossible that some continental species might recolonize Britain that way: The capacity of organisms to ensconce themselves in the world’s most inhospitable places—from lichens on Antarctic glaciers to sea worms in 176°F sea vents—may symbolize the meaning of life itself. Surely, as small, curious creatures like voles or the inevitable Norway rats slither down the Chunnel, some brash young wolf will follow their scent.
The Chunnel is a true wonder of our times, and, at a cost of \$21 billion, also the most expensive construction project ever conceived until China began damming several rivers at once. Protected by its buried bed of marl, it has one of the best chances of any human artifact to last millions of years, until continental drift finally pulls it apart or scrunches it like an accordion. While still intact, however, it may not remain functional. Its two terminals are just a few miles from their respective coasts. There’s little chance that the Folkestone, England, entrance, nearly 200 feet above current sea level, could be breached: the chalk cliffs that separate it from the English Channel would have to erode significantly. Far more likely is that ascending waters could enter the Coquelles, France, terminal, only about 16 feet above sea level on the Calais plain. If so, the Chunnel would not completely flood: the marl stratum it follows makes a mid-channel dip and then rises, so water would seek the lowest levels, leaving part of the chambers clear. Clear, but useless, even to daring migrating creatures. But when $21 billion was spent to create one of engineering’s greatest wonders, no one imagined that the oceans might rise up against us.
Condensation, alone, will block the tunnel at the bottom
Nevermind leakage. So long as the tunnel is open at both ends, wind will blow through. It will carry air that is warmer than the tunnel walls. Cold air can't hold as much water, so the water will condense out on the cold surfaces of the tunnel.
The continuous action of condensation will cause water to accumulate and run to the bottom of the tunnel. It won't be pumped out befause the tunnel has fallen to disrepair.
Keep in mind - the tunnel is a "V". It quickly descends to depth on each end, however, there is still a slight grade down to a "bottom point" - and that's done for drainage reasons - so water accumulates in one place, so they know where to put the pumps. Well, the French aren't so good at that, clearly.
Figure this will take a couple months til full blockage just from condensation, if seepage doesn't do it first.
Once it fills the bottom of a "vee", the wind will stop blowing and the condensation rate will slow greatly.