Is there any truth to the fact that an op-amp will behave ideally in closed-loop configuration only when it's closed-loop gain << open-loop gain?
Yes. However, modern op-amps have VERY high open-loop gain, which makes the error (usually) acceptably small. The caveat to this is that as your operating frequency goes up, the open-loop gain goes down.
A better way to express this is: an op-amp will behave ideally in closed-loop configuration so long as the open loop gain is >> than the closed-loop gain at the frequencies that you are operating at.
"Is there any truth to the fact that an op-amp will behave ideally in closed-loop configuration only when it's closed-loop gain << open-loop gain?"
To be "pedantic": No, there is no device that will "behave ideally". Of course, as outlined in the other answers, some closed-loop properties come very close to an ideal behaviour (input resistance in non-inv. configuration, output resistance, gain determined by external resistors only) - however, another important property even is degraded due to heavy feedback: Dynamic stability.
As a general rule, we must state that stability properties of closed-loop configuration have become worse if compared with open-loop conditions. This is quantitatively described by the phase margin (distance to the stability limit expressed in degree phase shift). As a result, the step response eventually will exhibit a remarkable overshoot or even decaying oscillations. There are some operational amplifier units (not unity-gain compensated) which are not allowed to operate for small closed-loop gains (heavy feedback) because they become unstable.
As a general rule: Negative feedback has many desired results and improvements - but also some drawbacks (dynamic stability). However, this demonstrates the classical situation in electronics: each design is a trade-off between several (sometimes) conflicting requirements.
For an ideal op-amp the gain is considered to be infinite and when we operate it in closed loop configuration,we do so to meet our requirements like how much gain we want for a articular circuit. So yes for an op-amp to behave ideally in a closed loop configuration the closed loop gain has to be very very less than the open loop gain.