The inputs to an oscilloscope are typically pretty well protected, and may be discrete transistors (especially older models). The problem is that there is a parasitic SCR/TRIAC structure in an IC that is not there in a discrete transistor.
In that case, the fault occurs when the IC is powered on with voltage already present at some of the pins, inputs or outputs. It isn't current drawn through (in or out) of a pin, it is current flowing from Vcc to ground through the substrate inside the IC die. It does not always destroy it if you shut it off again quickly, but it will likely get hot fast. Turning off power to the IC and removing the outside voltage source resets the intrinsic TRIAC.
It can also occur in a poorly protected IC when ESD (electrostatic discharge) hits a pin, input or output. It seems a lot less prevalent with newer ICs.http://www.fairchildsemi.com/an/AN/AN-600.pdfhttp://www.ti.com/lit/an/slya014a/slya014a.pdf
That is in addition to the problems Grumpy_Mike mentioned.