If $|\psi\rangle, U|\psi\rangle$ are known, how many pairs of such qubits are required to find the operator $U$?

Assume that we know a quantum state and the result of applying an unknown unitary $$U$$ on it. For example, if the quantum states are pure qubits, we know $$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$ and $$U|\psi\rangle=\gamma|0\rangle+\delta|1\rangle$$. Then how can we compute the unknown operator $$U$$?

If $$U$$ acts on a $$d$$-dimensional Hilbert space, then you need the result $$U|\psi\rangle$$ for a set of $$d$$ linearly independent vectors $$|\psi\rangle$$.
So, if you're talking about a single qubit unitary, you need two different states $$|\psi\rangle$$. If you have these, then by linearity you can find out $$U|0\rangle=\alpha|0\rangle+\beta|1\rangle\qquad U|1\rangle=\gamma|0\rangle+\delta|1\rangle.$$ Then, your $$U$$ is just $$U=\left(\begin{array}{cc} \alpha & \gamma \\ \beta & \delta \end{array}\right)$$ (the columns are just the individual outcomes for the computational basis states.)
You might wonder if you can get away with fewer. For example, we know that $$U|0\rangle$$ and $$U|1\rangle$$ must be orthogonal. However, there is still some freedom of parameters (such as a global phase) that you cannot determine, and hence you need to know both outcomes.