Is the distance between the Sun and the Earth increasing? $M$ = mass of the Sun
$m$ = mass of the Earth
$r$ = distance between the Earth and the Sun
The sun is converting mass into energy by nuclear fusion. 
$$F = \frac{GMm}{r^2} = \frac{mv^2}{r} \rightarrow r = \frac{GM}{v^2}$$
$$\Delta E = \Delta M c^2 = (M_{t} - M_{t+\Delta t}) c^2 \rightarrow \Delta M = \Delta E / c^2$$
$$\rightarrow \frac{\Delta r}{\Delta t} = \frac{G}{v^2 c^2}.\frac{\Delta E}{\Delta t}$$
Sun radiates $3.9 × 10^{26}~\mathrm W = \Delta E/\Delta t$
Velocity of the earth $v = 29.8 \mathrm{km/s}$
There is nothing that is stopping the earth from moving with the same velocity so for centripetal force to balance gravitational force $r$ must change. 
Is $r$ increasing? ($\Delta r/ \Delta t = 3.26070717 × 10^{-10} \mathrm{m/s} $)
 A: The Sun is also losing mass due to the solar wind. Again the fraction of mass lost is very small compared to the mass of the Sun, so the effect is very small.
There are these relevant papers that I think you will find interesting:
Orbital effects of Sun's mass loss and the Earth's fate
Astrometric Solar-System Anomalies
A: Such a small magnitude makes this process negliable among other factors; indeed this result literally means that you don't need to care about this process until you build extreamly accurate theory of the full solar system dynamics. To the extent that is probably unreachable due to deterministic chaos.
A: The distance is increasing due to friction from the tides. Besides the moon, the sun has a tidal component for the earths oceans, and when that crashes into continents the energy absorbed comes from the potential energy of the sun-earth system. I am not sure how this compares to the distance loss due to the mass and energy radiation of the sun you mentioned.
A: This is an old question, but I thought it might be worth chewing on a bit. The loss of mass due to fusion in the sun is piffle.  The Earth’s orbital radius will change more likely due to interactions with the other planets.  The first order perturbations in the orbital elements of the Earth are its eccentricity and right ascension.  The change in the orbital radius or the sem-major axis distance is higher order.  However, that can occur and there is an over all orbital drift in planetary orbits which is chaotic in nature.  The Earth is in a near 1/12 orbital resonance with Jupiter.  The Earth may over the next billion years shift away from this and enter into a near 1/11 orbital resonance with Jupiter, where our orbital radius is about 1.06AU.  
This early Earth may have been at .83AU relative to today’s orbit very early on.  This is an orbital resonance of about 16 with Jupiter.  The sun had a power output of 70% of current power.  If you factor these together you get a solar irradiance on the Earth comparable to today.  If the Earth had the same orbital radius as today, even factoring in a $\mathrm{CO_2}$ atmosphere temperatures would be $30~^\circ\mathrm C$ cooler than today.  Curiously if Earth does drift outwards this delays the solar death of the Earth.  If Earth remains at the current radius temperatures will become intolerant in 500 million years for complex life.  
Some numerical analyses of this I have run.  The interaction with Jupiter results in a periodic oscillation, and a computation over a longer period of time result in a drift which pushes the Earth outwards on average by about $4.2~\mathrm{km/sec}$.


$\bf[addendum]$
This is in part due to alpha Centuri’s commets.  One big uncertainty is with understanding the early Earth.  I did some homework on this and at 1AU about the warmest the Earth could have been is about -25C with various estimates.  Of course this is my interpretation of geo-modelling.
The orbital dynamics is based on computer modeling.  This is a general plot of 45,000 years.  I should have posted this image.  This illustrates the “signal” in these long runs, where the low frequency stuff has the largest amplitude.  This is the main signal for an outwards drift.

This does extend the future for life on Earth.  If this planet stays at 1AU the prognosis becomes grim about 500 million years from now.  The planet will start to reach temperatures 30C higher than today and complex life will begin to die out, and further in a billion years oceans will start to boil.  That will really foul things up.  However, with the outwards drift these time frames are almost doubled.  The luminosity increase in the sun will accelerate faster in time and over take this.  The outwards range on this is 2.5 billion years before the oceans start boiling.  Once the oceans start boiling this planet will transform into a 400C version of Venus.  So I figure complex life on this planet, life which emerged with the Cambrian revolution 550 million years ago, might have a good 750 to maybe 1000 million years ahead of it.
When I first read about the future time frame of life on Earth my mind instantly questioned what happened going back in time.  It implies a very cold early Earth; one where it seems the development of life would have been far more difficult.  
A: That is basically correct, however that change is not very significant. The orbit of the planets in the solar system is chaotic over long periods of time (2 - 230 million years according to this wikipedia entry), and this effect is relatively minor. Other causes for change of orbit include gravitational pull from other planets, collisions with asteroids, solar wind and other variables.
A: I think your reasoning is correct, but the values involved are very small.
In one year r will increase by 1mm, so in 1 billion years it will have increased by  1000 km or by about0.01%
A: Right now?  No, it's decreasing, until we get to perihelion (91.4M miles, near the January 3rd), and then it'll start increasing again 'til aphelion (94.5M miles, near July 4th).
(As Mark Eichenlaub pointed out -- Earth's orbit is not circular)
A: I think the reasoning has an error.  It assumes $v$ is constant, but instead we ought to assume the angular momentum is constant.
By dimensional analysis that leads to
$$r \propto \frac{L^2}{GM}$$
so as $M$ decreases, $r$ increases (the original post had $r \propto M$, not $r \propto 1/M$.
On the other hand, assuming a circular orbit seems dubious.  
As the other commenters said, this effect is minute.  A significant effect on the orbit of the moon around the earth is tidal evolution, which does actually push the moon further away.
A: I think even currently mass loss due to the solar wind is orders of magnitude greater than mass loss due to $E=mc^2$. But even so it isn't much. Ideally you would have enough
 mass loss, so that a planets radius would increase at just that rate that kept the stars
 luminosity divided by r squared constant, i.e. if that were the case the planet would stay 
within the habitable zone, even as the star brightened due to stellar evolution. Alas the
 solar wind is much too weak to accomplish the task. But we should be moving further out in
 any case.
A: Definitely Yes.
The Earth is moving away from the Sun at a rate ($0.57H_{0}$) that can not be explained by any of the current official models.
Once again I point to a MODEL in which the data is consistent with the theory. This model is public since 2002 (and my personal knowledge since 1982) in anticipation of the factual finding that the Earth was moving away from the Sun.
It is very problematic to place the origin of life in a cold world. With this model the life has started on an Earth full of energy.
I think that someday this model will be studied with due attention.
I am not the author of the model. I'm just a messenger (preaching in the desert?).
by G. A. Krasinsky and V. A. Brumberg, 2004
Secular Increase of astronomical unit from analysis of the major planet motions, and Its Interpretation

$\frac{dAU}{dt}=15\pm4\: m/cy$
at present there is no satisfactory
explanation of the detected secular
increase of AU

WeiJia Zhang, ZhengBin Li and Yang Lei, 2010
Experimental measurement of growth patterns on fossil corals: Secular variation in short distances ancient Earth-Sun

both the modern and ancient leaving
rates could be measured with high
precision, and it was found that the
Earth has been leaving the Sun over
the past 0.53 billion years. The
Earth’s semi-major axis was 146
million kilometers at the beginning of
the Phanerozoic Eon, equating to 97.6%
of its current value. Measured modern
leaving rates are 5–14 m/cy, whereas
the ancient rates were much higher.
Experimental results indicate a special expansion with an average
expansion coefficient of $0.57H_{0}$

A: The answer depends heavily on the time frame you are taking, and also on the definition of the distance between the Sun and the Earth. 
The Sun will expand after a while (a long while, but still...). And its surface will get way closer to the actual position of Earth. Will Earth still be near this position remains to be seen, but it is womehow likely, but not guaranteed. So the Sun and Earth will be closer together, but they will probably start by very slightly get further apart first...
To be honest, on the long run, there is no good and "correct answer", for the very fundamental reason that we are talking about a very chaotic system: the Solar System is a complexified version of the famous three body problem. Since we don't have enough information on the state of the System, we cannot predict its evolution. It may very well be that Earth, in a billion or 2  years, will be way way further or closer...
No equation, no clear answer... Sorry, but this question has, as of today, no clear answer.
A: Certainly the sun loses mass by radiation, and also by solar wind, but this is only one side of the story.  Mass falling into the sun must also be considered.  Comets routinely fall into the Sun, like one below:

Subsequent to this question being asked, E. V. Pitjeva published VALUES OF SOME ASTRONOMICAL PARAMETERS (AU, GM⊙, M⊙), THEIR POSSIBLE VARIATIONS FROM MODERN
OBSERVATIONS.., where $GM_{sun}$ was determined to be changing by less than 1 part in $10^{13}$ per year.   
