Daft question, no pun intended. If daffodil flowers are picked does it encourage growth and extend the flowering period?
No. They won't produce new flower buds this year. However, picking will have the same effect as dead heading by encouraging the bulbs to put all their foliage energy into making a big fat bulb for next year so the display gets better every year. You can help the process along by feeding with blood, fish and bone or pelleted chicken manure. Leave the foliage for at least 6 weeks or, better still, until it dies down completely.