Amplify my 3.3V DAC signal to 220V
A DAC and amplifier would be the expensive and complicated way to control the heater.

You have 230VAC at hand - you must, because that's what the heater operates on.

Use that 230VAC source, a few SSRs (solid state relays,) and a couple of diodes.

The control signal doesn't seem to drive the heating element directly.  The commands seem to be just that:  commands to a controller in the heater.  That means you don't have to switch high power, and can use small SSRs and diodes to make the control signals.

Take the simple cases first:  


Confort
Confort -1
Confort -2
Eco ou reduit


All four can be handled by one relay:


Relay open
Cycle relay 4 min 57 seconds off, 3 seconds on
Cycle relay 4 min 53 seconds off, 7 seconds on
Relay on


Each of the other two cases require a diode and relay each.  Put a diode in series with each relay, each diode points in a different direction.  

Now, you turn off the main relay, and turn on one of the two relays with a diode.  Each sends a different command 


Use zero crossing SSRs.  These switch when the AC voltage crosses zero volts.
Use SSRs rated for use on 230VAC.
Use diodes rated with a peak inverse voltage of more than 300V.
Use diodes rated to handle more current than the command circuit requires.


Doing that, you can use three digital outputs from a microprocessor to send all commands. 



Be very careful when working with 230VAC. There are many safety practices you should follow: look them up, learn them, apply them.  Failure to do so can kill you.

There are many design practices to follow when designing and laying out circuits that use 230VAC.  Look them up.  Learn them.  Follow them.  Failure to do so might burn down your house or electrocute you or someone using the device.