What's wrong with this circuit to convert 0-5V to +10/-10V
Your circuit is incorrect.  What you are trying to do is a gain+offset (negative offset) so you want to gain it up and offset the virtual ground point. You don't need that 10k/10k voltage divider as you've drawn it to provide a 2.5V reference. 

Since your full scale output from the DAC is 5V, and you want a full scale output from the opamp of 20V (-10 to +10) then you need to have a gain of 4 in the opamp. That gives you 0-20V. So you want (DAC x Gain) - Offset.

In a non-inverting op-amp, the + input has a gain set by the (feedback resistor/input resistor)+1.  However, the negative input has a gain of simply the (feedback resistor/input resistor).  So a non-inverting input with a gain of 4 will have a a gain of 3 from the negative input.  You can use the negative input to offset the fullscale 20V to -10 to +10.  So you want 10/3 = 3.33 volts at the negative input. 







simulate this circuit &ndash; Schematic created using CircuitLab

At the point I marked 3.333V, for a standard non-inverting op-amp configuration, that point would be GND. But you want to add the negative input's gain to the output too, so you need 3.333V there. 

The LM324 will work fine,but it's not a precision opamp and has some significant input offsets itself, and you might lose precision of the DAC input, but it can be trimmed out if you need to. If you don't want to trim anything then you can use a precision opamp with good DC characteristics.