amplifier for 10V LED dimmer control using Raspberry Pi
Rather than explain or fix the circuit you show which does not have a chance of working I will show what will work.

Specs:

0 to 3.3V source ( low Z) must be amplified to 0 to 10 out.
Load Impedance: assumed high   Z.        

Easiest solution is a rail to rail in &amp; out (RRIO) Op Amp (OA) with a 10V supply. (LDO)

Gain= 10/3.3=3.03 ( round up )  

Resistor range:  100k preferred.   







simulate this circuit &ndash; Schematic created using CircuitLab

Here is a discrete solution from 0.1V to ~10V







simulate this circuit