Why doesn't helium start to fuse while there is still hydrogen fuel? In all the descriptions of the stellar life cycle it seems as though helium doesn't start being fused until all (most?) of the hydrogen is gone.
Is this true? Why is this?
It seems counter intuitive.  Consider a fire consisting of gasoline (hydrogen), and wood (helium).  Sure the gasoline would burn up quicker, but the wood would still be consumed while the initial gasoline was being consumed.
 A: Temperature equates to the speed the nuclei are travelling at. Since Helium nuclei need to collide with greater energy to fuse this can only occur at a higher temperature.
As a gas gets hotter it expands and so becomes less dense, reducing the amount of energy generated by fusion.
With a mixture of Hydrogen and Helium the energy generated by Hydrogen fusion keeps the gas from becoming dense enough to fuse Helium, and once the energy from Hydrogen fusion falls enough, the gas can shrink enough to start fusing Helium. 
A: The analogy is facile. Helium fuses at a temperature ($10^8\ \text{K}$) roughly ten times higher than hydrogen ($10^7\ \text{K}$), so a better analogy would be alcohol and thermite. That higher temperature is achieved only by massive gravitational contraction after hydrogen fusion [EDIT: in the core] is exhausted.
EDIT:
To expand, different mass stars undergo radically different life cycles, so "hydrogen fusion is exhausted" means different things for different stars. In all cases, fusion occurs only in a tiny core region.
For the lightest stars, convection (think rapidly boiling water) churns the entire star, so all of their hydrogen will eventually fuse. This will take much longer than the age of the universe, but even in the distant future, they will never compress enough to generate helium-fusing temperatures.
For heavier stars, including the Sun, convection only mixes the core region, so exhausting hydrogen fusion only means fusing all the hydrogen in the core. The tricky wrinkle is that after the exhaustion of core hydrogen, after the core collapses and heats, and after helium fusion starts, a thin shell of previously-pristine hydrogen surrounding the fusing helium can, itself, commence fusing as well. Hence the first, clarifying edit above.
A: Actually hydrogen makes up the majority of the mass of a star throughout its entire life, even during helium fusion. Basically, things proceed like so:
Core hydrogen fusion produces the energy that powers the star for most of its lifetime.
At a certain point, all of the hydrogen in the core has been burned. Note that hydrogen is fusing in a shell around the helium core.
The helium core, since it isn't fusing -> not producing any outward force collapses inwards. As it collapses, it grows hotter and hotter, but there still isn't any helium fusion. The heat produced by this collapsing core actually causes the shell hydrogen fusion to increase like crazy. This extra hydrogen fusion causes the star to become a red giant.
At a certain point, the helium will become so hot and dense from its continued collapse, that it will finally begin to burn (~10^8 K as Andrew said.) This causes the helium core to expand, and the hydrogen fusion becomes confined to the core of the helium region. This actually cools the hydrogen shell by quite a bit, causing hydrogen shell fusion to decrease tremendously. This causes the entire core-shell output to drop, and the entire star becomes somewhat smaller as it collapses inwards. Now the star is running almost entirely on helium fusion.
This process is known as "helium flash."
But at the end, you are left with a start that is nearly entirely made of cooler hydrogen, with a small and extremely hot ball of helium at its core.
