A number of answers and comments have suggested that this would affect the axial tilt or orbit of Earth, so let's first work out the likely mass of the dragon to see how plausible those scenarios are.
The land area of Asia is $44.58 \cdot 10^{12} m^2$. (1) The continental crust is $30-50 km$ thick.(2) Density of the crust ranges from $2.2$ to $2.9\;g/cm^3$.(3) This suggests that Asia has a mass of about $2.94 \cdot 10^{21} kg$ to $6.46 \cdot 10^{21} kg$, which is between 1/1000 and 1/2000 of Earth's total mass. We'll take the larger number as a worst case scenario (although in practice, dragons are probably less dense than rock).
Now let's have the dragon fly up to the edge of the atmosphere, 100 km up (4). The total mass of the Earth-Dragon system is, of course, unchanged, so we just need to see how much the center of mass moves. Taking the old center of mass as our zero point, the dragon is a point mass of $M_{earth}/1000$ at 100 km + the earth's radius, or 6,456 km. The earth is a point mass of $999 \cdot M_{earth}/1000$ at -25km, since the Asia-shaped part we removed was all on one side. 921
Which gives us a center of mass 18.5 km away from where it used to be. The orbit won't care; even if the dragon flies away entirely, the orbital radius will change by less than 1 in 30 million; we can just add another leap day once every billion years or so and everything works out.
The center of mass of the remaining Earth has only moved by $0.1\%$, but it's no longer on the axis of rotation, so there will be a small but measurable change in axial tilt. Of course, when dealing with this much mass, small-but-measurable probably still means enough torque to cause the worst earthquakes in human history, but I'm afraid we've reached the limit of what I can calculate exactly.
The more immediate problem is that if the crust flies away, we now have an area of exposed mantle the size of Asia. The closest event in history to that would be the eruption that formed the Siberian Traps. Those are 7 times smaller than the exposed area (and the actual area that erupted would be smaller still), and that event is one of the more likely causes of the Permian-Triassic mass extinction, which wiped out 95% of all life, but arguably cleared the way for the rise of the dinosaurs. It's somewhat appropriate that this time, it's the rise of a giant reptile that leads to the extinction event, instead of the other way around. 921