If you look at the I/V characteristic curve for a Zener diode, there is a minimal Iz(min) at which the Zener/avalanche effect is engaged. However, this is not relevant for your clamping application. As an indication, Iz(min) is considered to be 5-10% of Iz(max).
Either:
a) the source that drives the input you want to clamp has enough current capability to engage the diode and thus the input will be clamped. The current will go up until the diode burns or the source reaches his limitations.
b) the source will not provide enough current (too much internal resistance) to reach the breakdown zone and the diode is almost linear, like a resistor. Forcefully, the voltage on the diode will be less than the breakdown voltage.
Cheers,
Dan