It is known that each year the Moon is moving away from the Earth due to tidal forces.
What about the Earth relative to the Sun?
Well, I wasn't sure if tidal forces between the Earth and Sun were strong enough to have any effect on the matter. Truth to be told, the article barrycarter linked clarifies that tidal forces have neglible effect relative to the Sun's nuclear fusion mass loss.
Some more details on this.
Tidal "tugging" is a two step process. The Moon's gravity raises a tide on the Earth and this tidal bulge, spins ahead of the moon and pulls the moon very slightly faster, which slowly pushes the Moon away from the earth. (Note, if the Earth rotated much more slowly, the effect would be the opposite, the orbiting moon would be ahead of the tidal bump and gradually slow down).
When the Moon was closer and the tidal bulge was a lot bigger and the corresponding push away was significantly faster. Still not fast, but quite a bit faster than the current 1.5 inches per year, perhaps several feet a year after formation if I was to guess. Even more for the brief period when the Earth had an entirely liquid-magma surface, but that was probably pretty temporary.
So, most of the distance the Moon moved away from the Earth after formation happened in the first billion years. The move away speed decreases with the distance, something roughly equal to the 4th power of the distance.
As you said, this exact same thing happens between the Earth and Sun, but the effect is even smaller, so small that it's not the primary effect in changes in the Earth's average distance from the sun.
Tides depend on gravitational field variation (not field strength), but on how much the field changes near side of object to far side. It also helps if the larger object that's being orbited has a liquid surface, oceans or molten or a large ocean under an icy surface. If the Earth didn't have oceans the Moon would moved away from the Earth much more slowly and it would be quite a bit closer to the earth than it is now.
Rocky surfaces still bulge a little so the effect still happens but it's smaller.
The sun's is plasma which is very fluid so the Earth and all the planets create a tidal bulge on the sun, but from the sun's point of view, all the planets are quite small, just tiny dots in the sky, so the tidal effect from the planets creates only a tiny tidal bulge on the sun.
Tidal effect can be roughly compared to the size of the object in the sky. for example, the Moon and Sun from the point of view of the Earth are about the same size in the sky, so they have similar tidal effects. The Moon's larger tides is due to the Moon having greater density than the Sun. More on Tides here.
(this doesn't apply to a tennis-ball in your hand, however, which can appear the same size as the moon or sun, because you have to consider the size from the center of the earth, not the size to your eye on the surface.)
The tidal force the sun receives from the Earth from 93 million miles away when the Earth is just a tiny dot, so effect and corresponding the tidal bulge is quite small, so the corresponding push is very small.
A curious sidebar, is that when the Sun goes Red Giant, the larger sun will have a greater distance variation to the Earth, which will probably be the closest planet at that point. The tidal bulge will be correspondingly larger and the Earth, quite a bit closer to the bulge, so when that happens, the tidal push the Earth gets will be measurably larger. Which as Rob Jeffries' correction points out, could slowly draw the Earth into the Sun.
A point to add, when the Sun goes Red giant and loses mass, as Red Giants do, then the expansion of the Earth's orbit due to mass loss could be pretty significant. I don't have the skill to even try to run those numbers though.
The tug from other planets, while, even smaller, can have an effect in moving planets too, more so when the planets are in orbital resonance, which, currently, no planets are. The solar system passing through a dust cloud can have a slowing down effect too. Needless to say, all the effects are quite small.
Yes - the earth and sun do have tidal forces like the moon and earth. There are two main reasons this is happening.
The sun is always losing mass due to nuclear reactions, the sun is always converting a tiny amount of its mass into energy. This means the pull on the earth is weakened.
The tidal forces that happen between the earth and moon also happen between the sun and earth, except on a much smaller scale.
The sun is also losing its angular momentum at a rate of 0.00003 seconds per year
The theorized reason I saw on either the NASA or Discovery Channel for the Moon's orbit spiraling away from earth is that it's formation resulted from a collision between the Earth and an asteroid, and the resulting momentum carries it away.