How can I determine the voltage the resistive load in this relay?
The typical IRF540 \$I_D\$-\$V_{ds}\$ relationship is shown in this chart from the IRF540 datasheet:



This shows that with 10 V \$V_{gs}\$, which is about what you could expect to be applied with your circuit, the typical IRF540 will be in triode operation (what we often call "fully switched") with up to about 100 A load.

That means you can essentially treat the MOSFET channel as a resistor when it's in the on state and conducting less than 100 A. The datasheet also guarantees a channel resistance of no more than 77 mΩ with 10 V \$V_{gs}\$.

Now the current through your load will be 

$$I_{L} = \frac{24\ {\rm V}}{R_L + 77\ {\rm m\Omega}}$$

again, provided the current works out to less than about 100 A.

If the circuit is well-chosen for your load, it will mean you get darn near 24 V across the load.