Basically what you are asking for is called a digital to analog converter, or D/A or DAC for short. In this case you want the full range to be 0-10 volts.
From your description, it appears the receiving end passively pulls up the line, and is expecting the dimmer to put a variable resistance between it and ground. You want to outright control the voltage, but you only need a low side active pulldown to do it. Here is a circuit that will probably work:
The input to this analog circuit is a 0 to 3.3 volt digital PWM signal from a microcontroller. R5, C4, R2, and C2 form a two-pole low pass filter that makes the average value of the PWM signal. Since your frequency requirements are so low, you can easily create such a PWM signal in a microcontroller with plenty of resolution. For example, a 1 kHz PWM signal will have its PWM frequency reduced by nearly 4000 (over 70 dB) by this filter. Even slow micros can give you 8 bits or more resolution at 1 kHz PWM frequency. The micro would adjust its PWM duty cycle in response to commands received via a UART or some other digital interface.
The opamp is used in the classic positive gain configuration, except that since the transistor inverts the signal the opamp inputs are flipped in response. R1 and R4 form the feedback divider, which in this case causes the circuit to have a gain a little over 3. Ideally you want a gain of 3.03, but the values shown give you a little bit at the top of the range where you know the output will go to maximum. The opamp drives the base current of Q1 to whatever it takes to make the desired dimmer line output voltage. R3 is there so that there will be some voltage change in the opamp output with output change. Otherwise, the opamp output would always be at the B-E junction drop above ground, which could lead to instability. You didn't say what the maximum current is that a dimmer has to sink. This circuit can handle well over 100 mA, which is probably high. If so, you can make R3 higher, but the 1 kΩ shown should work anyway.
C3 is there only for stability. You don't need much bandwidth, so there is no harm in overdamping the opamp. Some capacitance here will be needed since even with R3 there, there will actually be a voltage gain less than 1 from the input to the opamp output.
The previous circuit accidentally had the opamp inputs flipped. The transistor inverts the voltage, so the opamp inputs have to be opposite from the usual positive gain configuration. The circuit above is now the fixed version.
I have also updated the circuit for the processor running at 3.3 V instead of 5 V and now show the PWM signal from the micro directly.