To understand the problem associated with a Digital Storage Oscilloscope (DSO) at low sampling rates, it's important to grasp the basic concept of sampling and Nyquist's theorem. Nyquist's theorem states that to accurately represent a signal, the sampling rate must be at least twice the highest frequency present in the signal. This minimum required sampling rate is known as the Nyquist rate.

However, if a DSO samples a signal at a rate lower than the Nyquist rate, aliasing occurs. Aliasing is a phenomenon where the high-frequency components of the signal are misrepresented as lower frequency components, often resulting in the displayed waveform appearing distorted or incorrect.

Let's evaluate each option:

• The entire display screen will permanently turn off due to over-sampling: This is incorrect. Over-sampling does not cause the display to turn off.
• The input signal will automatically switch to a direct current measurement mode: This is incorrect. The DSO does not switch modes automatically due to sampling rate issues.
• The analog input probe tips will melt due to excessive heat generation: This is incorrect. Melting of probe tips is not caused by sampling rate issues.
• Aliasing error, where the displayed waveform appears as an incorrect low-frequency signal shape: This is correct. Aliasing results from undersampling, causing the signal to appear at a lower frequency.

Therefore, the correct answer is:

Aliasing error, where the displayed waveform appears as an incorrect low-frequency signal shape.

In conclusion, when using a DSO, ensure that the sampling rate is at least twice the highest frequency of the signal to avoid aliasing. This fundamental criterion guarantees accurate signal reproduction and avoids misleading visual interpretations.