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 $CO_2$ atmosphere temperatures would be $30C$ 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 $4.2km/sec$.
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.