controlling SPI devices using the Arduino - different voltages
It does matter. The MCP4922 has a input voltage limit of VDD + 0.3v on any input pins. At 3.3v power, anything above 3.6v will probably kill the chip. And the Arduino ATMega has a Input Voltage Level High cutoff of VDD * 0.6. At 5v Arduino power, that means it needs to see at least 5v * 0.6 = 3v on a input pin to properly register as a logic HIGH. So this should work one way before you need to add things.

Sparkfun has a nice tutorial on how to get it to work between two chips at 5v and 3.3v, but they all require extra components (some resistors or diodes, or a mosfet). It also goes more in depth about the voltage levels.

But the most important thing is why power the ADC at a different voltage than the Arduino, if you have the 5v source available? Is this just a hypothetical?