simulate this circuit – Schematic created using CircuitLab
Figure 1. The domestic supply has one wire 'neutralized' by connecting it to earth either at the local supply transformer, at the property entrance or both, depending on local regulations. Any current flowing in the neutral lines will cause a voltage rise on the neutral conductor due to cable resistance.
If I stuck one multimeter test lead into ground or neutral on a live power strip with some things plugged in, then connected the other to e.g. a radiator or something metal that might be connected to the physical ground, would the meter show a voltage?
The voltage will be given by \$ V = IR \$ where I is the current returning in the neutral and R is the neutral resistance. In practice it's more complicated because there will be other devices returning current to the neutral and these will add to the voltage.
To keep voltages close to specification (115 V or 230 V) the cable resistance is kept low so that voltage drop doesn't become a problem and N-E voltages should only be a couple of volts. Daisy-chain a few light-guage extension leads together to power a heavy load and it does become a problem as neutral voltage could be > 10 V.
Would anything short circuit?
No. Multimeters are high impedance (AC resistance) when switched to voltage measurement. Only a tiny current would flow. Don't try switching to current range when doing this as you would provide a low-impedance alternate path back to the transformer via the pipework and ground.
Remember that if the neutral conductor breaks between your test point and the supply that your neutral will go live if anything is plugged in (and switched on).
simulate this circuit
Figure 2. The neutral has broken and so the right-hand side of the neutral wire is connected to L via LAMP1 and could provide a lethal shock.
Never assume that it's safe to touch a neutral.