If you use the double screen, the impact would be minimal and perhaps even boost your honey yield with a larger foraging population from having two queens laying for a period of time.
I would leave the old queen on the bottom with plenty of room to lay and supers for nectar. Followed by the double screen board with a few frames of capped brood, honey, pollen and the new queen. Once the new queen starts laying and your happy with her performance, you can then dispose of the old queen and remove the screen board.
If you use a push in cage, you need to dispose of the old queen first. You will then have a gap in brood rearing until the new queen gets laying. It probably isn't that big of deal in the big picture. I prefer the double screen method because you always have the old queen still there to fall back on if you have issues with the new queen. You can also make a nuc with the old queen as well until your happy with the new queen, but that can be a little more work.
Here is how I use the double screen board.http://robo.bushkillfarms.com/beekeeping/queen-introduction/