Question: Ion thrusters require a lot of energy because of their high exhaust velocities. But why can't they just pump a mixture of hydrogen and xenon into the chamber, mix it with oxygen and burn it? Doesn't this ionize the gas and also provide enough thermal heat to push the plasma through the grids?

Maybe not hydrogen and oxygen, but some other exothermic reaction. Not to completely avoid electricity input but reduce it. Maybe reducing the size of the solar panels. I am talking about deep space missions.

This would also be much more efficient than energy production through turbines, wouldn't it?

Is there any drawback of using exothermic reactions?

Answer: 
The ion thruster should be as simple as possible and it should use the propellant mass effectively. Why use three gases if using xenon only works? The propellant mass should be accelerated totally, not only the xenon part. If you use the full heat of a reaction of hydrogen with oxygen, the combustion chamber must be cooled to prevent melting. An ion thruster is used for long missions due to the low thrust, but liquid hydrogen and oxygen can't be stored that long. High pressure tanks for gaseous hydrogen and oxygen are heavy. Xenon is a very heavy gas, a tank for the same mass of gas could be much smaller and lighter for xenon than for hydrogen or oxygen. The amount of electrical energy used for the heating and ionization of xenon is much smaller than that used for the acceleration of xenon. You don't need additional consumables to heat the xenon, just a part of the solar panels is enough.

Chemical energy could only deliver about 2 % of the energy that an electric accelerated ion gets.
Because what you are just describing is regular liquid fueled rocket.

The efficiency of a rocket fuel is dictated by its exhaust velocity.
This velocity has a maximum value for each kind of fuel.

If you are going to burn hydrogen, you can achieve about $4\ 400 \frac{\text{m}}{\text{s}}$.
Ion thrusters reach about $29\ 000 \frac{\text{m}}{\text{s}}$

Moreover, we cannot store liquid hydrogen/oxygen for a long time. For a deep space mission, this is an issue.