How to interface Beaglebone with a temperature condtioner with 0-10V as output?
The LM317 is not what you are looking for. It's a voltage regulator, it does not translate a voltage level, but it does generate a stable output voltage, even with variable input.

What you need is voltage level translation, as provided by e.g. a level shifter or a voltage divider.


Voltage divider


simple
needs careful dimensioning to avoid problems
in case of BBB needs precautions like a Zener diode to clamp the input voltage to never exceed 1.8 Volt.

Level shifter


more complex
safer
output voltage range can be set by feeding in V_ADC
Needs a reference voltage for the input side





For controlling relays with the BBB you can use its GPIO functionality. Note though that you can NOT drive relays directly from the GPIO pins as their current consumption generally exceeds the pin drive capability of the AM335x SoC on the BBB! You must either use relay modules with built in drivers (transistor or better an opto-coupler) or attach a transistor/optocoupler (module) between each GPIO pin and relay.



Feel free to ask for clarification or further pointers.