[Some of what I say below was already covered by JohnnyBerg, since he responded as I was writing this up]
With the first drawing, it looked like you were trying to get to 0V output with the -1.25 volt reference summing into the DAC output.
Was that a goal? because the second drawing, without the LM317 and just using the pass transistor is much better, but it won't go down to 0V.
You'll need the opamp to drive the transistor's base below 0V to get 0V out, since the emitter is always 1 V
be higher than the base. I'm assuming it's an NPN pass transistor.
2) you might need to use a darlington pair to get the drive strength you need, depending on the current requirements and the beta of the pass transistor.
3) with or without the darlington pair, you will need to remember that OPAMP U3 must drive it's output higher than the desired output voltage. Some jellybean , low cost opamps are not rail-to-rail outputs, so the opamp output can only reach a volt or 2 below the supply rail. i.e. if the supply is 18V, then the opamp output can get to only 16V, and then the max output can be 1 V
be below that, or 2 V
be's if you use a darlington pair. This limits the output to V
supply - 2 - (2 V
be's ). To be safe, say the V
be = 1V at high current, so the output will be < V
supply - 4 = 20-4 = 16. So your output goal of 0-15V is achievable.
4) You can reduce the drive requirements of the opamp U3 by using an NPN driver and a composite NPN-PNP for pass transistor (I think this was first described by Bob Pease but I might be wrong):
(image edited with corrections)
In this configuration, the opamp does not need to go to high voltage a few V
be's above the output as JohhnyBerg and I said. This configuration drives the first NPN Q101 in a small range of about 0-1V that allows the opamp to be run off +5V. In this way, the high-voltage rail can now go much higher than 20V (as long as the pass transistors can handle the voltage), and you don't need to worry about the op-amp running at the same high voltage as the output rail. You can run the opamp at a more reasonable voltage. Note that the 2 diodes on the emitter of Q101 actually move the range up by 2 diode drops, or about 1.4V. So the transistor operates at about 1.4V to 2.4 V at the base, and this means the OpAmp does not need to go to 0V to get 0V at the pass transistor's emitter (the output of the PSU) . This solves the 0 volt problem without needing a negative supply.