Is the length of the day increasing?

The length of a day is increasing slowly. The rate is very slow, about 0.0017 seconds per century.

The length of SI day is based upon the mean solar day between 1750 and 1892. A mean day nowadays (!) is about 0.002 seconds longer, which means that we accumulate a difference of about 0.6 seconds every year. This is why we have leap seconds (leap second is when we stop our clocks for one second to let the earth "catch up"). From the Wikipedia article on leap seconds:

The leap second adjustment (which is approximately 0.6 seconds per year) is necessary because of the difference between the length of the SI day (based on the mean solar day between 1750 and 1892) and the length of the current mean solar day (which is about 0.002 seconds longer). The difference between these two will increase with time, but only by 0.0017 seconds per century. In other words, the adjustment is required because we have decoupled the definition of the second from the current rotational period of the Earth. The actual rotational period varies due to unpredictable factors such as the motion of mass within Earth, and has to be observed rather than computed.

Also see: ΔT.

The Moon moves away about four centimeters a year on Earth, 15 while the Earth's rotation is slowing down, which will in the distant future total solar eclipses occur stop the moon not having sufficient size to cover the solar disk.

In theory, this separation should continue until the Moon takes 47 days to complete one orbit around our planet at which our planet would take 47 days to complete one rotation around its axis, similar to what happens in the Pluto-Charon system.

An associated effect is that the tides slow the Earth in its rotation (energy lost due to friction of the oceans with the sea), and since the Earth-Moon system has to conserve angular momentum, Moon makes up away now, 38 mm each year, as demonstrated by the laser distance measurements, made possible by the retro-reflectors that the astronauts left on the moon.