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: 
Ion engines need the ions to stay ionized
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To be usefully efficient, an ion engine needs to ionize a substantial fraction of the propellant, and keep it ionized until it passes through most of the positive electrostatic potential gradient to whatever potential the local free space is assigned. I won't say "ground" here, but the assignment of zero is arbitrary. 

It's the electrostatic acceleration of the ions that is the "push" of the engine.

That means the plasma needs to be low enough pressure so that the ions and electrons don't recombine too much. In some designs using electron cyclotron resonance, the vacuum must be so low that the electrons can accelerate to high energy before a useful, ionizing collision. That low pressure is way too low to allow a sustained chemical reaction. 

Stuff doesn't burn in a near-vacuum
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An ion engine's biggest job is to keep generating high energy electrons to continue to ionize the neutral gas constantly introduced into the chamber. There are then so few collisions that the reaction of one oxygen/hydrogen atom pair wouldn't trigger any nearby atoms to undergo a similar reaction. It wouldn't be self-sustaining, and in fact almost all the hydrogen and oxygen would pass out of the engine unreacted.

[![enter image description here](https://i.stack.imgur.com/C8TFY.jpg)](https://i.stack.imgur.com/C8TFY.jpg)

**above:** from this YouTube Video [Introduction to Ion Thrusters](https://www.youtube.com/watch?v=grU8g9jnS4w). Right-click and open in a new view for full size readability and detail. Image is placed here to avoid link rot if the YouTube video is ever moved or deleted.

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Here is a link to the actual video:
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.