My '01 V70 T5 is doing the same thing. Starts then stops immediately. No error lights are showing. Let me know how you fixed yours, please!
If I unplug the MAF sensor, then the car will start but it runs rough...and of course that triggered a code to "Check Engine."

 

Probably the dumbest answer you have, but I have to make sure.

Any chance that you are using the "service" key that doesn't have the chip in it? My 99 S70 does the exact same thing if I'm not using the Normal factory key.

 

No, I tried both ignition keys and even turned them 180 degrees but no go.

 

Excuse my necroposting, but we sold the S70 over a year ago.

However, as of yesterday, I am a proud owner of a red 2001 Volvo S60.