Before I say anything I just wanted to say that I really hate the word "paradox". Because people think it means "something which is self-contradictory". But in actuality the way the word is used, especially in mathematics, it means something which is really crazy and surprising. For example, consider the Birthday Paradox. The paradox is that it is more likely to find two people with the same birthday in a group of 23 random people than not. There is nothing contradictory about this statement. It is just surprising. The Monty Hall Paradox (here) is not some contradiction, it is a puzzle that for some reason confuses a lot of people. The only paradox that I can actually think of at this moment that has a self-contradictory nature is the Russel Paradox. All other paradoxes are just surprising statements. I guess the misunderstanding of the world "paradox" comes from the misapplication of the English meaning of "paradox" to math and science. Paradox has a literary meaning as well, it means that the character of the story does something which is contradictory to what he does (something like that). So the confusion over what paradox means in math and science is confused with its meaning in English literary interpretations. Anyway, let me return back to what I wanted to say.
Olber's Paradox says that the universe must be finite. If the universe was infinite then there would be infinitely many stars in the sky. If there would be infinitely many stars in the sky then the entire sky would be brightly lit up. But instead we see the stars "discretely", meaning, we see a point there, another dot there, and so forth. The sky, when we look at it, is basically all dark with a few bright dots in the sky. Therefore, the universe must be finite.
There are a number of problems with this paradox. The first obvious problem is that the universe can be infinite with finitely many stars. Maybe the universe keeps on going forever but it has finitely many stars? But there are more serious problems with this paradox, even assuming there are infinitely many stars.
The first serious problem with this paradox is more of a scientific objection. That is, people cannot see objects from an arbitrary far distance. The weaker the light source the less of a distance one can see the source. For example, I think that in a extremely dark valley one can see a candle from 30 miles away. But if the candle is too far away then the person would no longer see it. His eyes are not strong enough for collecting such weak signals of light. Indeed, this is the case when one observes the heavens with his eye and when one observes the heavens with a telescope. With your own eye you can see about 6000 stars (if you live outside a large city where the sky has no interference with anything). With a telescope you can see a million. Under a telescope the entire sky brights up. Therefore, we only see a finite number of stars in the sky. There could be an infinite number of stars, but we only see finitely many. Therefore, there is no paradox.
The second serious problem is even assuming there is an infinite number of stars and even assuming we have perfect vision it is still possible for there to be gaps in the sky. It is still possible for there to be an infinite universe, with infinitely many stars, with a perfect vision, and we still see an almost empty sky. To see how this is possible consider the xy-grid. Let the origin (0,0) represent you. Now cover this grid with points (n,m) where n and m are integers. These integers represent the stars in the sky. If you join (0,0), where you at, with a star at (n,m), you get a line. This line is the line of vision. If you remember from high-school math the slope of the line between (0,0) and (n,m) is m/n (assuming n is non-zero). But m/n is a rational number because n and m are integers. This means that all the vision lines have rational slope. If you remind yourself a bit more high-school math the equation of a line passing through the origin at (0,0) with slope r is given by the equation y=rx. This line is a line of vision with slope r. Now if r is a rational number then it will hit one of the points (n,m). Meaning if you look at the night sky with rational slope you will see a star. But if r is irrational then the line y=rx will fail to hit any of the stars at (n,m). This means this line is a clear line of vision i.e. it does not see any star in its path. So we see from this simple example that it is possible for there to be infinitely many stars, and even under the assumption that we have perfect vision, we can still find lines of vision that do not see any star. Of course, this argument is two dimensional, but we can just as easily extend this argument to three dimensions, we just worked in two dimensions because it is easier to follow the argument.
But the biggest problem with this paradox is that we can prove that no matter how we position infinitely many stars in this sky there will always be clear lines of vision. This argument is a little more advanced because it no longer uses high-school math. It uses the important concept of "countable" and "uncountable", that was introduced by the mathematician Georg Cantor at the beginning of the 20th century (which is among the most important ideas of modern mathematics). Think of the xy-grid again. On this grid randomly position the stars anywhere, with (0,0) representing us. The number of these stars, represented as grid points, will be a countable number. Which is infinite, but it is a certain size of infinity. The number of all lines passing through the origin, which have the form y=rx, where r is any number, is also infinite. But it is a different kind of infinity. This kind of infinity is uncountable. The infinity of all lines (representing our line of vision) is a larger kind of infinite than the infinity of all stars. Therefore, there will always have to be lines, an uncountable number in fact, that would be clear lines of vision. And so we will always see gaps in the night sky.
Olber's Paradox is utter fail.