Well sure, it would not be a perpetuum mobile. It would take CO2 from the air to turn into fuel, and once it runs out, there is no more fuel to be made. It creates a little CO2 by burning fuel for itself, but if that is less than the amount used for fuel production (that's the scenario where it could power itself), that would be less than it needs to generate the fuel to continue the cycle.

An oil pump can pump enough oil to fuel itself (we don't need solar to power our gasoline industry). And we can eat a plant and have enough energy to cultivate more plants to eat. I would not be surprised if it works the same here.

Edit: I guess the flaw in the logic is that, where there is a lot of energy stored in oil and plants (and plants get it from soil and the sun or so, again something external), this process is the reverse of burning. So if reverse burning costs less fuel than burning, we'd have a perpetuum mobile. Is that it? I'm sorry that this wasn't immediately obvious to me (if that is even the answer), since there are so many other fuel creating and self supporting processes...