Is the distance from the Earth to the Sun increasing, and if so, by how much in kilometers per (Earth) year?
First I should say that the Earth's orbit around the Sun is elliptical, not perfectly circular, so the Earth-Sun distance is changing as we speak just from the Earth traveling in its orbit around the Sun. See here for a discussion of that.
Is the orbit itself changing? Well, there are some long-period oscillations, but those are very small, and don't imply that we're systematically moving towards or away from the Sun.
There is an effect which is making us move very slowly away from the Sun. That is the tidal interaction between the Sun and the Earth. This slows down the rotation of the Sun, and pushes the Earth farther away from the Sun. You can read about tides, as they relate to the Earth-Moon system here. The principle for the Sun-Earth system should be the same. But how big of an effect is this? It turns out that the yearly increase in the distance between the Earth and the Sun from this effect is only about one micrometer (a millionth of a meter, or a ten thousandth of a centimeter). So this is a very tiny effect.
There is another effect which is also small, but somewhat bigger than the tidal effect. The Sun is powered by nuclear fusion, which means the Sun is continuously transforming a small part of its mass into energy. As the mass of the Sun goes down, our orbit gets proportionally bigger. However, over the entire main sequence lifetime of the Sun (about 10 billion years), the Sun will only lose about 0.1% of its mass, which means that the Earth should move out by just ~150,000 km (small compared to the total Earth-Sun distance of ~150,000,000 km). If we assume that the Sun's rate of nuclear fusion today is the same as the average rate over those 10 billion years (a bold assumption, but it should give us a rough idea of the answer), then we're moving away from the Sun at the rate of ~1.5 cm (less than an inch) per year. I probably don't even need to mention that this is so small that we don't have to worry about freezing.