A lot of things happen, depending from what perspective you look at it. I'll first start with some scaling laws. Note that
~ means 'scales with' or 'is proportional to'.
Basic scaling laws:
Lets start with:
(1) length ~ length
(2) density ~ 1 (is constant)
From (1) we can derive that:
(3) area ~ length^2
(4) volume ~ length^3
From (2) and (4) we can derive that:
(5) mass ~ length^3
We also know that accelerations on an object don't change (example: gravity):
(6) acceleration ~ 1
force = mass * acceleration, from(5) and (6):
(7) force ~ length^3
acceleration is contant (and here it may get a little weird):
(8) time ~ sqrt(length)
From (1) and (8):
(9) velocity ~ sqrt(length)
The idea from time may more comprehensible when you look at vibrating stick: a very long stick will have a longer oscillation period than a short one.
Now for the aerodynamics. As the length scale increases, to keep everything the same (proportional) the plane will also have to fly faster (by the sqrt of the length scale). This can also be seen from the lift equation:
L = C_L * 0.5 * rho * V^2 * S
L = W, for which holds that W and L (forces) ~ length^3
length^3 ~ 1 (from C_L) * 1 (from 0.5) * 1 (from rho (density)) * V^2 * length^2
V^2 ~ length
V ~ sqrt(length)
The increase in both length and velocity increase something called the Reynoldsnumber, which is equal to:
Re = (rho*V*x)/mu where:
rho = density of medium (air)
V = velocity
x = characteristic distance
mu = viscosity of the medium
As the Reynoldsnumber increase, in general the drag gets lower for a given amount of lift. This also allows (for most airfoils) a larger maximum lift coëfficiënt (the lift per area generated at a certain velocity and density).
Also your Mach number increases, since velocity increases.
M = V/a where
V is the velocity of the aircraft
a is the speed of sound
As M > 0.3 you have to take this effect into account. It decreases your lift coëfficiënt. However 0.3M is around 100 m/s, and your craft probably won't fly at that speed. At 0.3M the error is about 5%.
These are the 2 main aerodynamic factors that come into play. Note that both numbers are dimensionless.
You can also think of other factors such as the range on your transmitter, structural changes in proportions. Controlability: as time increases with length, your craft will feel less agile, requiring longer approaches which may be out of sight.
But I seemed you wanted to know mainly about the aerodynamics. With the basic scaling laws, you will be able to derive what happens with most other factors you encounter.