LEDs in parallel - How does this circuit behave?
Each of the six series strings of LEDs is configured to draw about 20mA, for a total current through R5 of 120mA. If one LED burns out then the total current will drop by about 20mA. The current drawn by the remaining five strings will increase a bit, but you should still see a significant change in the voltage across R5.

You linked to the wrong datasheet for the ULN2003. If you are using an actual Darlington transistor then you need a resistor in series with the base, about 1k\$\Omega\$ should be good. Look up \$V_{CE(sat)}\$, the collector-to-emitter saturation voltage for the Darlington transistor. This is the voltage that appears across the transistor when the Arduino output is high. You can assume that this voltage won't change much if one string burns out, so the voltage at the junction of R4 and R5 is what you want to watch (as @jsotola mentioned).