Can the increase in thickness of a piece of paper which can be folded infinitely surpass the speed of light?
No, you can't, for a couple of different reasons.
The first is the difficulty of folding paper more than a few times. Mythbusters managed to fold one sheet 11 times I think, using a very large sheet of paper and the help of a steamroller. It took a lot longer than 5 seconds per fold.
The second issue is more fundamental. You could resolve the first issue by just cutting the paper in half and stacking one half on top of the other instead of folding it. But then you have another problem: suppose your stack of paper has reached one light year in height. Next you have to cut it in half and put one half on top of the other to make a two light-year stack.
With some cleverness you can do the cutting as quickly as you want. (For example, you could cut it using a carefully timed laser pulse from far away.) But once it's cut you have to move one half of the stack upward by one light year, so that the bottom of that half lines up with the top of the other half. You can't move the stack faster than light, so no matter how you do this it has to take at least one year. The next iteration will take two years, the next four, and so on, and the top of the combined stack will never move faster than light.
So really the logic of your question has to be reversed: it's not that you can move faster than light if you fold a piece of paper every five seconds, it's that you can't fold a piece of paper indefinitely every five seconds, because doing so would mean moving something faster than light.
(There is a third issue too, which is that every time you cut the paper in half you reduce its size, and eventually you'll just have a stack of atoms that you can't cut. But of course you can always just start with a bigger sheet of paper.)
As David Starkey points out in a comment, you can actually do a factor of two better than this, if you don't mind the bottom of the stack moving as well as the top. Then you can move one half of the stack down at the same time as moving the other up, so each one only has to move half a light year instead of one. But of course this doesn't change the overall argument. Each end of the stack is still limited by the speed of light, so you can't double the height of a one light-year stack in less than 0.5 years.
It is a fact of nature that nothing moves faster than c, the velocity of light in vacuum.
One can imagine infinite scenaria for things moving faster than light, simpler than folding a paper. A continuous acceleration in vacuum is the simplest.
It is an observational fact that nothing moves faster than light. Innumerable observations of particle physics and astrophysics have not falsified this statement..