No.

This is a misconception propagated by the media. It's not anything new; it actually existed long before the LHC started operating, and made many people scared for no reason at all.

If someone tells you that the LHC is dangerous because it is going to create black holes, then they don't know what they're talking about, and you shouldn't listen to them.

So here's what's really going on:

The LHC is an experiment designed to look for new particles and other undiscovered physical phenomena. One of these is the hypothetical existence of extra dimensions.

"Extra dimensions" sounds very science-fictiony, but it doesn't mean "parallel dimensions of reality" or anything like that, it just means there might be additional spatial dimensions in addition to the 4 dimensions that we know of: 3 spatial dimensions (length, width and height) and 1 time dimension.

If extra dimensions do exist, then they are very hard to detect, which is why we haven't seen them yet. However, one proposed method of detection is as follows.

Assuming that the extra dimensions have some specific properties, then it might be possible that if you put enough energy in one place you will get some effect due to the extra dimensions which you will then be able to detect.

These effects come from the hypothetical possibility of a "quantum black hole" (or "micro black hole"). This is a very very very very very tiny black hole, tinier than a proton, which will exist only for a very very very very very tiny fraction of a second and then immediately evaporate.

Since it's so tiny and evaporates so quickly, we won't even see this quantum black hole. The only thing we'll see is some particles that were left after the black hole evaporated.

It's important to realize that these hypothetical quantum black holes are not the kind that can grow. In order to create an actual, stable, black hole you would need an insanely large amount of energy that no technology that exists now or will exist in the foreseeable future will even come close to.

With the tiny and almost insignificant amount of energy that the LHC has, we can never hope to create stable black holes even in our wildest dreams!

So what we can do? If extra dimensions actually exist, and if they have suitable properties, then we might fantasize about maybe creating a really really tiny quantum black hole that will exist for an extremely short duration of time and then immediately disappear.

We are not anywhere close to creating stable black holes that can grow in size, and we will not be anywhere close to it in this century or the next.

So you can stop worrying... :)