Fridge Horror / Inferred Holocaust: The world of the future has no disease or microbes. But, the Time-Traveler must have brought microbes from the past and the Eloi and Morlocks were pawing him all the time, so...well, remember what happened to the Martians?
For all we know the reason the Eloi and the Morlocks have no disease is the result of genetic tinkering or immunization which would make them immune to all diseases, past and future. Smallpox may be almost wiped out, but we still immunize against it.
Fridge Logic - The titular Time Machine moves in Time, but not in Space. So, if it is constantly occupying the same space, even though it is moving quickly in time, if it is sitting in the same space, then rules of physics dictate that no two objects can occupy the same space at the same time. Since the time machine exists in all time it travels through, and maintains the same space, then shouldn't it be visible to all who pass by, or at the very least, the space incapable of being occupied by anything else in that time?
The story claims it goes through time too fast to be seen, comparing the effect to a wheel spinning so rapidly that the spokes become invisible. This doesn't solve the problem of it not being solid, however. If you reach through a spinning wheel, the spokes will hit your hand whether you can see them or not.
Well one way to explain it, consistent with the temperature changes the traveller experiences, would be to say that, for example the time machine experiences one microsecond every hour - if you happen to be looking at the right place at the right time you might catch a flash of it, but will probably not notice it. However, for the guy in the 'time bubble' each microsecond of contact with the normal world, if the normal world around you is a glacier, will cause temperature decrease.
Better yet, if he doesn't move through space while the earth itself continues circling the sun, he should have landed in the vacuum of space pretty soon.
Clearly he does move in space then, in a manner consistent with the Earth's gravitational field. It's a lot easier to explain how to avoid this problem with a 'static' time machine which gradually moves through time than e.g. a Delorean which travels instantaneously.
A novella published in Analog in the 1980s addressed these points (the following is a paraphrase, maybe somebody could find the story to correct the quote): "All time-travel stories have assumed that the traveller becomes intangible. They have also assumed that the traveller is affected by Earth's gravity and thus doesn't go floating off into space. Put these two effects together and you don't have The Time Machine, you have Journey to the Center of the Earth."
Saying that rules of physics dictate that no two objects can occupy the same space at the same time is more than incorrect, as you don't know the physics behind time travel and they may introduce a new quantum number which changes depending on whether particles are traveling in time normally (100% speed forward) or not. Rules of physics dictate that no two fermionic particles can have the same set of quantum numbers - with that new one introduced, it's not a problem. And it doesn't even matter to bosons anyway. Also, it may modify interactions, which could cause the time traveller to stay in the same place on Earth without disturbing things that move through him. Also, relativity is all about that you cannot point to any absolute point in space. Any kind of movement must derive form the (modified by the new quantum number) forces, like gravity. It won't just stay in some absolutely defined place (which doesn't exist) when the Earth flies away. Then it all makes sense. You can't say any time travel physics are bad, because they work in experiment (the movie universe) and you can't give the good ones. Intuition is not science.
In other words, the time traveler doesn't end up in space for the same reason most other things don't. Velocity is relative to something we arbitrarily say is zero, like a point on Earth's surface. There's no such thing as absolute zero velocity. If you stand still, you're still moving 30 kilometers PER SECOND relative to the sun, but you don't go flying off into space because Earth is moving at that same velocity. Same deal with the time machine.
I don't mean to pile on to anyone, but I'd like to add to the "what about space" question which is often brought up in discussions of Time Travel. Given that not only the Earth, but the sun, the rest of the galaxy, and the local group of galaxies are all constantly moving — relative to other celestial objects — just what should a time machine be "still" relative to? "Earth" is just as good an answer as anything else. To put it another way, it's not just about the fact that velocity is relative, but location is, too. There are no absolute coordinates in space, no "center point" in the universe where the coordinates are 0, 0, 0.
I think it can be best put thusly: According to Newton, yes everything is in motion, but there still is some absolute framework of space and time that motion can be compared to. But Einstein came along and showed that, no, there is no absolute space at all. It's not that we can't measure it, it doesn't physically exist. There is, however, a four-dimensional block of absolute space-time with events sprinkled within it. We see space as three-dimensional "cross-section" of this block with our motion dictating how the "cross-section" is sliced. If we change our motion, and we change how the block is sliced (an excellent illustration here◊) So, for our time machine, saying that travelling through time would mean it would leave Earth and end up in space is not necessary justified by the physics. It's possible that it would remain on its original path and stay on Earth.
Never mind not being able to see the time machine — I can't believe that over the course of 800,000 years nobody walked through the space that the time machine occupied, or built a wall across it.
Maybe it became a "haunted spot" where something invisible instantly annihilates anything that is inserted in that space, like an object moving at relativistic speeds would destroy anything it came in contact with?