How much more life could the Sun acquire via star lifting? It has been proposed before that we could use this technique to remove hydrogen from the Sun to lower it's rate of fusion and extend it's life, so it doesn't fry our planet.

I am wondering how much more life could we possibly get out of the Sun doing this? I am thinking we will want to keep the Suns luminosity constant, perhaps lower than it is now even, it used to be only 70% but perhaps that is too low without us having to have too much carbon dioxide in the atmosphere for comfort (1000 ppm is the limit before we start getting affected iirc). 

Anyway, in order to to that I believe the star lifting would have to be done regularly over the future lifespan of the sun so a very long term project. It has to be regularly tuned so to speak as the core depletes it's fuel supply and collapses a little, we will have to remove a little more material to restore balance to keep the same power output.

Does anyone know how to figure out how long we could keep doing this? 

There will then of course, come a time when we can't remove more fuel from the sun because the core will have exhausted all accessible hydrogen by then and will have to resort to helium fusion. 

Or can we keep the suns output constant even with helium fusion? Or do we by then have to consider radically different methods, like somehow removing the spent fuel from the core and replacing it with the hydrogen we removed the previous billion(s?) of years prior?
TL;DR: the main sequence lifespan of the sun can be increased by a factor of 12.2.

Perhaps the most complete astrophysical analysis of stellar engineering for extending Earth habitability is Martin Beech's book Rejuvenating the Sun and Avoiding Other Global Catastrophes (2008). 

In order to maintain the biosphere the sun's interior need to be mixed (in order to avoid the red giant phase) and material needs to be removed (in order to avoid a too great luminosity).

Material removal

Removing material is tricky because the potential energy at the Sun's surface is $U_\odot = GM_\odot/R_\odot=$ 1.9e11 J/kg. While just 2e-6 of the mass-energy embodied in the matter and 0.3% of the potential fusion energy it is still cumbersome. If a Dyson sphere was used at perfect efficiency the maximal mass extraction rate would be \begin{equation}
\dot{m}=\frac{L_\odot}{U_\odot} = \frac{L_\odot R_\odot}{GM_\odot} \approx 2.0061\cdot 10^{15} \text{kg/s},
\end{equation}
a typical mass for a minor asteroid. The timescale to disassemble the sun using just solar luminosity is \begin{equation}\tau_{disassemble}=\frac{3GM_\odot^2}{5R_\odot L_\odot} = 18.852\text{Myr}.\end{equation} Long before this time changes in pressure would reduce luminosity, making this estimate an underestimate by about an order of magnitude. I get 556.93 Myr, a bit more than Criswell's 300 Myr. 

Another idealized model is that the lifting process fuses hydrogen (using the $p-p$ process) to power further lifting. In this case each kilogram of lifted mass produces $6.45\cdot 10^{14}$ J, allowing lifting of 3380 kg more fuel. This could of course speed up arbitrarily if there was technology for it.

I am leaving out how to do it. Beside Criswell's classical proposals (speeding up rotation, sucking off plasma using magnetic fields etc.) Beech suggests a stream of ramscoops or a stellar wrap could do it. This is not really key to the question. 

Luminosity decrease

Luminosity can also be decreased by increasing non-thermal core pressure, perhaps through strong magnetic fields or rapid internal rotation. This would extend the main sequence lifetime as \begin{equation}\tau_{NTPS}\sim \tau_{MS} \left(1-\frac{p_{NT}}{p_T}\right )^{-7.5}.\end{equation} A 10% increase over current pressure would double the sun's main sequence lifetime and about halve the luminosity at hydrogen exhaustion. Achieving this is nontrivial given the current solar structure (the magnetic fields are generated in the convective outer envelope), but Beech notes that were the mass reduced below about half the current mass the modified sun would be full convective and magnetic fields would also produce a higher $p_{NT}$ in the core.

Another way of reducing luminosity is increasing the stellar opacity $\kappa$. In order to reduce the luminosity by a factor of 100 (and the surface temperature by a factor of 10) the opacity needs to be increased by a factor of 100. However, Beech notes that merely adding heavy elements will not help since they increase mass and hence luminosity. While one can envision "dialysis" where helium is removed and replaced by larger $Z$ elements this appears complex, and there are no local readily available sources of enough high $Z$ material in the solar system.

Mixing

Engineering enhanced mixing before the mass is removed is nontrivial. Impactors and lasers simply do not penetrate deep enough to matter. Beech suggests employing a sub-stellar mass black hole oscillating through the sun to mix it. If the hole is small, the time until absorption is long.  

Result

With mass-loss and complete mixing Beech finds the main sequence lifetime increased up to a factor of 12.2, with reasonable factors between 4 and 6.

Beech's scenario has the problem that the late stage sun while having an acceptable luminosity at Earth's orbit will be far hotter, releasing significant UV radiation. The end result is in many ways similar to a blue straggler star.