I've heard that a larger wheel (diameter) in the front can produce quicker turns on a motorcycle. How much larger should a front wheel be than a rear wheel on a road bike (if at all)? Will this affect the odometer/speedometer readings? Gas mileage?
A larger front wheel is more stable than a smaller one, so theoretically should keep you more stable in high speed turns. But, you'll be messing up the front fork geometry (rake and trail) which may have unforeseen consequences with the suspension and steering.
You'll definitely have more trouble in parking lots.
Dirt bikes and enduros do this. Choppers too. Their frames and suspension styles are built around the wheel sizes though. So, unless you are riding a bike that comes "stock" with two different sized wheels, I would not recommend it. On any bike that goes highway speed, you should be especially careful, since you may place more strain on forks or other components than they were tested under due to geometry.
Best to stick with the sizes that are on there.
If you insist on changing the diameter of your wheels, note that the speedo drive is on the hub and shouldn't change your speedo accuracy. You may experience slower acceleration with a larger rear wheel. You will experience reduced braking and reduced overall performance the larger your wheels get in general.
A larger wheel diameter WILL influence the dashboard's readings IF the sensor that reads data is placed on the wheel (like on bicycles, but usually on motorcycles it is placed near the brake), as it calibrated for a certain wheel. If the sensor is placed (very often) on the front sprocket, readings won't be affected.