Does an aircraft's nose landing gear extend on take off?
No, the gear doesn't actively extend, and any passive change to the strut extension during takeoff is likely to be imperceptible.

What you are perceiving during takeoff is the somatogravic illusion. It is a physiological sensation, not a physical change. Rapid acceleration, as during a takeoff roll, causes the inner ear to report what your brain assumes is your head tilting back.

The otolith organs in your inner ears sit in fluid and detect the tilt of your head. Acceleration and pitching up (or tilting back) cause the exact same movement of the otoliths. Because the otoliths are pushed backwards by the fluid, just like they would by you tilting your head back, it feels like a climb. 

When an airliner is taking off, you have limited outside references to the horizon and your inner ear takes over the task of orienting you. You might notice it in smaller aircraft if you closed your eyes; in very small aircraft, the acceleration may be too slow to trigger the illusion.