Are there currently observed systems of close (narrow) binary brown dwarfs? All over the Universe, there are (gravitationally bound) binary star systems. Considering the huge number of brown dwarfs, are there currently known double systems of gravitationally bound binary brown dwarfs? Given their comparatively low gravitation, how long would they need to melt together to a larger object?
What happens if binary brown dwarfs collide? Is this anyway a matter of any interest?
You've got a lot of questions here, so I'll try to answer them one by one.


  Are there currently known double systems of gravitationally bound binary brown dwarfs?


Certainly there are. I don't study low-mass stars so I'm not up on the literature enough to know any good examples off the top of my head. Kozaky suggested the Luhman 16 system in his comment and a quick google search also brought up the triple star system Epsilon Indi with a central K type star being orbited (at ~1460 AU) by a binary brown dwarf system. Perhaps some other more knowledgeable person can suggest more systems of interest.


  Given their comparatively low gravitation, how long would they need to melt together to a larger object?


That depends on a variety of factors, including their respective masses and initial distance apart. As two objects orbit one another, they continually release gravitational waves due to their interacting gravitational fields. These waves carry away energy and to conserve energy, the stars must lose angular momentum, thus getting closer. The time necessary for two stars of mass $m_1$ and $m_2$, with an initial radius $r$ to fall into each other is given by

$$t = \frac{5}{256} \frac{c^5}{G^3} \frac{r^4}{(m_1m_2)(m_1+m_2)}$$

where $c$ is the speed of light and $G$ is the gravitational constant. We can apply some basic numbers and calculate an order of magnitude time. If I assume $r = 1\:AU=1.5\times10^{11}\:m$ and $m_1 = m_2 = 2\times10^{28}\:kg$ (10 times the mass of Jupiter) then I find the timescale for collision into one object is

$$t = 5\times10^{30}\:s \approx 100\:billion\:trillion\:years$$

This is obviously way longer than the age of the Universe and effectively means that two Brown Dwarfs will never collapse. This fact could have been seen from the equation. Notice that the mass of the objects is in the denominator. That means that with lower mass means longer timescales and the fact that the distance is to the fourth power means the distance is a much stronger controlling factor here. You need huge masses to offset that $r^4$ for you to get a collision on reasonable timescales. Even doing a best case scenario of $r = 0.1\:AU$ and $m_1=m_2=10^9\:kg$ still gives me $\sim100\:thousand\:trillion\:years$.


  What happens if binary brown dwarfs collide?


Not much? I don't know if anyone has ever really tried to work out the theoretical work for such a problem. Given the answer above, the only chance two brown dwarfs are going to collide is by a freak chance that they're on a collision course in space and not in a slowly degrading orbit, but the chances of that are pretty minuscule as well.

One situation that may be of related interest is to consider regions of star formation. While it is somewhat contested, some people believe stars can form "bottom up". That means smaller objects form first, then those coalesce into larger objects and on upwards until you end up with big, bright stars. In this scheme, you would first form many many brown dwarfs in a tight area and these would dynamically interact and collide with each other to form larger stars. This isn't the same scenario of two lone binary brown dwarfs colliding, but its the best you're going to get if you want to see colliding brown dwarfs.

In this scenario, I think that since brown dwarfs aren't fusing and their main energy output comes from core collapse (by some Kelvin-Helmhotz mechanism), it isn't going to be a hugely explosive result (comparatively) to smash two together. They will of course burst out with energy in the process, but it won't be as dramatic as two fully-fledged stars merging. And that's probably a good thing. If you had a star forming region with many brown dwarfs that were slowly coalescing into larger stars, you wouldn't want each merger to blow out huge energies as it would prevent further stars from coalescing.


  Is this anyway a matter of any interest?


From an academic standpoint, I'm sure it is. As I said, I don't study this topic, but understanding the affects of brown dwarf collisions would be very useful for fleshing out star forming theories. I'm sure one could think of numerous other academic benefits for studying such a topic.

From a practical standpoint, I can't say understanding brown dwarf collisions immediately benefits humanity. But as with any topic in science, you never know how knowledge gained today may help in the future.