Well I guess my question is then how do you encourage intrinsic motivation in a task that the child is not interested in doing. Howler mentioned music lessons, for example. Another example: homework/ "boring" schoolwork.
To my thinking, the most direct way is to draw attention, in an age and temperament appropriate manner, to natural consequences of doing vs not doing X. I think Dude and others alluded to this. This may mean letting your child fail, provided the consequences aren't too disastrous. Behaviours that are truly important/have meaningful consequences in either direction will tend to elicit a motivating effect.
On the flip side of the coin, I think there is also value in learning when *not* to push forward in low urgency and importance tasks. If a 1/2% assignment is totally valueless and occupies a full day to complete, I can see an argument being made for not doing the assignment at all and substituting an interesting personal interest project instead. Life is simply too precious to waste on nonsense.
I'll give a trite example:
My DS16mo refuses to sit for his bath. He's too young to reason with in a sophisticated way, so I bought a bath mat and rubber faucet guard and join him, either in the tub or with my hands next to him. When he gets overly boisterous, I let him do a controlled fall, which scares him enough to buy some more reasonable play for another few weeks. He doesn't get hurt, I'm not over taxed, and he still has the freedom to enjoy baths in his own preferred way. Great. The global issue is safety, not sitting.