Should full flaps be deployed on takeoff?
First, you have to remember on some configurations, the lower stages of flap will mostly add drag and very little lift. Sometimes that extra notch of full flaps is only there to change the camber of the wing to add a bit of a nose down attitude to help with visibility on landing. Both are items you do not require on takeoff.

Now, when selecting flaps for take-off there are two things to consider: runway length and obstacles to clear after the runway:

Runway Length

Generally when lowering flaps your $V_{\text{mu}}$ (the speed at which you become airborne, “minimum unstick speed”) will decrease. This  means a shorter runway for normal stages of flaps. As you continue to add more flaps your $V_{\text{mu}}$ will be lower, but because of the added drag, which decreases acceleration,  it will actually take a longer distance to reach that lower $V_{\text{mu}}$. There is a sweet spot in-between, though. “Take-off Flaps” is not always designed to give you the shortest takeoff roll.

Obstacle Clearance

And that is because once airborne, the more flaps you have deployed, the higher the drag, the less excess thrust available, thus the less you can climb. Simply put, you are now flying at a low speed and barely climbing. Not a good idea if there are tall trees waiting for you at the end of the runway.

It would be illegal  to take-off on full flaps if:


POH says it is prohibited,or
For commercial flying, if the flap setting does not provide the minimum required gross obstacle clearance climb gradient (depending on the type of operation and number of engines, between 2.4%-3.0% ).