The voltage actually dropped by the primary of a transformer is set by the number of primary turns and the magnetic flux developed in the core. The lower the magnetic flux in the core, the lower the voltage on the primary side.
With normal voltage transformer, the core is sized so that with the full primary voltage across the primary coil, some relatively small magnetizing current will flow (relative is a relative term; this magnetizing current will be 10% to 50% of the full load current of the device!). But the point is that with the core not saturated, sufficient magnetic flux is produced to significantly impede the supply voltage.
As secondary current is drawn, it has the effect of 'balancing' some of the primary current, reducing the core flux. More primary current flows to keep this balanced, and we get the happy result that as we draw secondary current we see a corresponding increase in primary current flow.
In a current transformer, the core is sized rather differently. Remember that the primary is in series with the load, so that even if the primary impedance was _zero_ the current would still be reasonably limited. In fact, it is desirable to have a low primary side impedance for a CT, so that it doesn't waste energy that would otherwise go to the load.
In normal operation, there is very little secondary impedance, so current flows in the secondary to match the current flowing in the primary. Rather than seeing greater current flow in the primary, the result is to reduce the primary side impedance. Current flows in the primary, current flows in the secondary, and very little magnetic flux is actually produced.
If you open the secondary, then the current flow in the secondary ceases. Now, if this happened to be an extremely large transformer with enough core to produce lots of magnetic flux, then the open secondary would increase the primary side impedance. A large fraction of the supply voltage would be dropped across the primary, and the voltage delivered to the load would drop quite a bit.
But this is a small current transformer, which means that there isn't much core area, and thus not too much magnetic flux. For most applications, the load in series with the primary would still be the dominant control of current. Based on the current and core area, you could calculate the magnetic flux in the donut, and from this you could calculate the primary and secondary voltage drops. The primary voltage drop will be much less than the full supply voltage, but the secondary voltage drop will still be 80x the primary drop.
I thought iwire answered fungguy's original question very well, but it's always educational to delve a little deeper into theory.
If I may summarize:
1. The secondary current in a CT is a fixed fraction of the primary current. As Larry pointed out, 1/80 is common.
2. Ideally, the meter and wiring on the secondary side have minimal impedance. That way, even though several amps may be flowing, the voltage is negligible and the wasted power is minimized.
3. The minimized impedance on the secondary is also "reflected" into the primary, so the voltage drop is negligible.
We return now to the discussion of minutiae:
Remember that the primary is in series with the load, so that even if the primary impedance was _zero_ the current would still be reasonably limited.
I didn't understand that statement, but I agree with everything else.
As to the voltage present at the open secondary of a CT, I just don't believe the "hot-stick" levels being suggested. That voltage will be a function of the secondary turns count and the magnetic flux magnitude, and nothing else. As Jon mentioned, CTs have relatively small cores that are easily saturated, so the flux magnitude won't be nearly high enough to induce multi-kV voltages on the secondary.
Just how much voltage can be dropped across that big wire or bus passing through the CT with the open secondary? I'd be surprised to see more than a few tenths of a volt, so the secondary voltage can only be 80 times that, on the order of a few tens of volts.
What will happen is that the saturated core will dissipate a lot of heat, which certainly could damage the windings. That's probably the real reason never to leave the secondary open.