How long until the sun cannot sustain human life on earth? The sun will last, at its current brightness for 9 billion more years. How long until the sun gets burned down to the point where it cannot sustain life on Earth anymore?
Updated: I am more concerned with how long until human life cannot be sustained, not how long until the water on Earth vaporizes.
 A: Stellar Life cycle: We know that sun and other stars generate heat and radiation through nuclear fusion. Generally, there are several stages that a star would've to cross through its life so called the Stellar Evolution.
Our Sun (1 solar mass) is a Mid-sized Main sequence (specifically G-Type or yellow-dwarf) star. Nuclear fusion takes place between protons to form helium at this stage (Reactions are provided here). It's surface temperature is about some 5000 °C and core temperature is about $10^7 K$. As more and more hydrogen fuel is consumed, the layers of hydrogen are decreased simultaneously while expanding the shells filled with helium. Finally, the core begins to contract still expanding the helium layers and by this time, the core would've reached $10^8 K$ which is strong enough for the fusion of helium to produce carbon and by this way, the Triple-alpha process begins. During the expansion of Sun, we'd arrive at a time (about 3 billion years) when the temperature in Earth could no longer hold water and all lives would've been terminated. Also, The inner planets Mercury, Venus and Earth would've been engulfed by the expanding sun.
Wiki has a good comparison for Red Giant... Future of Earth article is also good.
A White dwarf: Once the triple-alpha process in a red giant is complete, stars weighing less than 4 solar masses (like our sun) don't have enough energy to ignite the carbon fusion process and then explode into a Supernova. So, The star collapses to a white dwarf until it's halted by the pressure arising from electron degeneracy. The Hertzsprung-Russell Diagram is a plot of luminosity vs temperature which shows the decrease in luminosity of stars with decrease in their temperature. A teaspoon of a white dwarf would weigh 5 tons. A white dwarf with one solar mass would be about the size of the Earth.

It has been estimated that it takes about 5 billion years for Sun to become a Red Giant. But, now we have much larger situations to concern at present like protecting ourselves from PHA's (Potentially Hazardous  Asteroids) than thinking about near future. One thing to keep in mind is that, More brainy guys are out there to think of building something like the AXIOM in Wall-E..! So, humans have chances of survival if those physicists build some crazy items to be placed in our future inventory list.
This paper explains the survival of a planet named V391 Pegasi b from a Red Giant (but only as a roasted one..!) 
Edit: Then, You don't require any of the above facties... The answer is: "How long would you stay in a desert..?". At about 1 billion years - Sun would be 10% more luminous so that organisms start dying due to Moist greenhouse effects (Water starvation). At about 1.6 billion years, Sun's luminosity is at about 40% and hence all Life on Earth dies due to Runaway Greenhouse effect..!
A: As for your question of "when will this happen" i think prior poster who answered misunderstood.yes the current estimate is about 7.6 billion years until the sun reaches its maximum but what i think he was asking is when will the earth no longer be able to support life due to the increased luminosity of the sun.
the answer to that question would depend on what studies and who you believe but the current range is appx. 1B to 1.6B years until the surface of the earth is no longer inhabitable by life as we know it right now.this point im referring to is the point water on the earth vaporizes and all the water based life currently on the planet dies if any is still left by then and hasnt been roasted , such as life living at the bottom of the oceans that may be able to ride it out right to the bitter end.
if life continues on earth from that point it wont be water based life forms like us unless we manage to construct artificial city like enclosures under the earth to protect ourselves from the heat and radiation.much more likely by that time we'll have figured out how to terra form mars and we'll be living there trying to figure out how to hop to the next planet , moon , or maybe even asteroid and continue our branching out into the universe.this is all assuming we dont blow ourselves up tommorrow in a nuclear war of course.its also possible , and if we still exist quite probable , there will be artificial life on the planet at that time.cyborgs , robots , whatever you want to call them , harvesting the last of the earths minerals we are capable of extracting from our soon to be dead world.
so really it is all in how you look at it.hope this helped.
A: If the Sun becomes a giant star,The Earth will not be able to support life. Since the giant stars have large radius; therefore when the Sun becomes a giant star, it may become so large as to engulf Earth, in which case the planet will be destroyed. Even if this does not happen, the sun will expand so far out that the temperatures on Earth will become extremely high so that all oceans will evaporate away, and there will be no water left on Earth. 
