A changing magnetic field induces emf in a circuit which is placed in it. This induced emf causes current to flow through it. This is given by Faraday's law which states that induced emf results due to rate of change of magnetic flux.
Current flows only through a closed circuit. Thus, when a wire is moved up and down in a magnetic field, it will not induce emf because the wire does not form a loop. It is an open circuit.