evolution - Is better healthcare a bane to the long-term survival of the human race?

The theory of natural selection has it that individuals with better genes tend to survive and reproduce, passing their genes to their offspring.

Yes, under selective pressure, that is. The stronger the selective pressure, the more the population will change. Without a selective pressure, there isn't an impetus to change.

However, due to the advancement in medical science, humans with poorer genes tend to survive and reproduce just as well.

You're making a mistake. "Poor" and "Good" are purely contextual. You might say that Sickle Cell Anemia is a poor lot to be stuck with... Unless, of course, you're in an area ravaged by Malaria. Then it's a boon.

What would be more accurate is that there aren't any strong forces acting upon humans that causes selective pressure and the proceeding adaptation. Of course, we are the result of such evolution, so by a basic standard all modern humans who reproduce are fit in a Biological sense.

For example, in the past, many people, excluding those with natural immunity due to some genetic mutations, would have succumbed to illnesses such as malaria and typhoid. But with better hygiene and medical treatment, these patients tend to survive.

The brain is the one of the finest products of evolution. Because we're no longer dying en masse at the whims of bacteria or virii (usually) does not mean we haven't adapted. Our ability to build advanced tools and perform science is an adaptation. One for which we pay heavily; we're practically helpless until our teens.

Now, imagine a future where all illnesses including cancer can be treated. How would it impact the survival of the human race?

We'd live longer, and assuming the rate of reproduction stays the same, there'd be a lot more of us.

Would we become more and more vulnerable such that our survival hinges heavily on medical technology, analogous to how an astronaut's survival is dependent on his spacesuit?

Well... Vulnerable to what? To disease? Probably not. Current Immunizations and treatments often utilize the body's own immune system; basically getting sick and all the benefits of encountering the disease without the side-effects of significant death.

To genetic diseases? No. The human population has exploded in the last hundred or so years, and our genetic diversity has been greater than at any point in the last 40,000 years when a major bottlenecking event occurred. For a genetic disease to deal significant damage on a species, it would have to be suffuse throughout the population. That's not happening.

To a post-apocalyptic world where technology breaks down and suddenly we're facing off against parasites and predators? Well, if the event ever occurs, then a lot of us won't be ready, that's for sure. The stress such a life put on us in the past limited our lifespan to nearly half what it is now, but unless there's a complete lack of knowledge many basic safety habits should persist. Cooking, boiling water, dressing wounds, how illnesses spread - these are all things which dramatically increase quality of life when utilized correctly.

Will we ever reach a point where we, quite literally, cannot live without technological intervention? No idea. Humans have very, very strong reactions to things which aren't human but appear human (The Uncanny Valley is a great example). It's just as possible that those instincts will prevent us from such a fate as the possibility that we'll be completely fine with becoming cyborgs. Nobody can even begin to give you a time-frame, though. As-is, there's no reason to become cyborgs (no Selective Pressure), and won't be for the foreseeable future.