Electrical – Does this neon indicator light require a resistor to limit current

If you want a simple solution, you could look for "current sense amplifier" ICs, which are designed specifically for this purpose.

The approach you describe is using the op-amp as a comparator, which is a viable strategy, but it will behave unpredictably when the input voltage difference is around 0V, so somehow you need to introduce an offset in the comparison.

There is a sample current sense circuit in figure 27 of this application note. The quick way to analyze that circuit is to use the rule of thumb that an op-amp in a negative feedback configuration (meaning with a connection from the output back to the \$-\$ terminal) will make the voltages at its inputs equal. I can explain in more detail if it would be useful.

To create a current threshold detector from that circuit, you could add a comparator on \$V_{OUT}\$, or you could remove the \$R_3\$ / \$R_5\$ divider and connect the \$-\$ terminal directly to the low side of the sense resistor \$R_1\$. The \$R_2\$ / \$R_4\$ divider would then set the trip point of the detector.

You can sense whether current is flowing through a circuit using a very small-valued (fraction of an ohm) resistor which, in such a role, is called "current-sensing resistor". When current flows, a small voltage develops across the resistor. Not enough to produce any appreciable voltage drop in series with the device, and not enough I^2R to generate any serious heat. This small voltage can be amplified (e.g. by an op-amp) and then something can be done with that signal, like feeding it to an LED-driving comparator circuit.