When you have input signals that exceed the closed loop bandwidth of your amplifier, the opamp is basically operating open-loop. The gain will be limited, unpredictable, and have large phase shifts. In the limit where you exceed the open-loop gain, the amplifier does nothing at all, and the transfer function is basically governed by whatever gets through the passive network. This is particularly troublesome in active filters. An active low-pass filter may have insufficient attenuation at high frequency due to the absence of feedback. Furthermore, the op-amp relies on feedback to present a high impedance. At high frequency, the input simply looks like a diode. This can cause nonlinear distortion. This is a common cause of GSM interference in audio circuits. If the input does not sufficiently filter RF, it can get rectified by the op-amp input, forming a crude AM radio. GSM generates lots of RF bursts, which demodulated cause a lot of noise. This is how a transmitter at 1.8 GHz can interfere with an audio signal at 1 kHz. It is best to use passive filters to eliminate any signals beyond the bandwidth of your op-amp.
There is a second problem with high gain and oscillation, but this problem is generic, not specific to op-amps or operating above their rated bandwidth. The problem is stray coupling from the output back to the input, which can cause oscillation if the phase is such that you have positive feedback. With high gain amplifiers, the forward gain is so high it only takes a small amount of feedback to oscillate. The feedback can come from several sources. but typically gets worse with increasing frequency. This is a circuit layout problem, and the way to avoid this is to split your gain into multiple stages, and to keep the high level output physically separated and/or shielded from the low level input. Don't make a U shaped amplifier chain with a volt output next to a nanovolt input...