Operating an adjustable voltage regulator below minimum Vout-Vin
Let's have a look at datasheet Figure 10 "Dropout Voltage". At 500mA it will be a bit above 1.5V.

When operating in this mode,  the output pass transistor will be fully on (saturated in this case, since it is a NPN bipolar).

If this was a PNP pass device LDO, you would expect excess ground current, as the regulator attemps to saturate its output device. However, this one uses a NPN, so the excess base current will simply go into the output. No problem here.

The regulator will not regulate anything though, this means it will act either as a resistor or as a couple of diodes in series, so output voltage may vary depending on current draw. Also, output voltage will follow input voltage, so input noise will not be suppressed.

If your load works on 19V, and you have 24V input, and the load can tolerate the regulator not rejecting input noise, then you're fine. In this case it would simply act as a voltage limiter. 

If you also want to filter noise, then something like a capacitance multiplier with an output voltage limit would be more suitable.

EDIT: Example







simulate this circuit &ndash; Schematic created using CircuitLab

This is a simple capacitance multiplier. It lowpass-filters the input (RC network) to filter out noise. Zener limits the voltage. I put in a CFP (double transistor) for lower output impedance, so you can say that's the "luxury" version!