First, we are talking physics so we have to be accurate. An EMF will only be induced in a wire if it passes through a changing magnetic field.
It is often helpful to relate physical phenomena with what you can directly see. In this case, we can use a mechanical analogy to explain what happens.
The mechanical analogue of EMF is force and the mechanical analogue of current is motion or velocity. When you push on an object two things could happen--It could move or you would just be exerting force on it. The same thing happens in this case: if an EMF is generated and there is enough conductivity in the circuit a measurable current will flow. Compare this with pushing an object such as a cart, which has wheels that makes it easy to resist the friction on ground and respond to the force applied.
Even if the conductivity is low(an open circuit is in fact completed by air surrounding the wire), a current will flow, but it will not be significant. This is similar to pushing a wall; it won't move significantly.