A PWM output is going to 0V if I connect an ADC input in a development board
Figure 1. Original circuit.
R1 is not wired as a potentiometer. It's wired as a variable resistor or 'rheostat'. If you move the wiper to the right the op-amp will see 5 V. If you move it to the left the op-amp will see \$ \frac {200k}{20k + 200k}5 = 4.55 \ \text V \$.
Meanwhile your op-amp has a gain of \$ 1 + \frac {R_4}{R_3} = 1 + \frac 1 {3.7} = 1.27 \$ so at minimum input voltage the output is trying to get to 5.77 V but, fortunately, I suspect, is limited by U1's supply voltage. Since you're only getting 3 V maximum it probably means you are not using a rail to rail output op-amp. Check the datasheet to see what the maximum output voltage swing is.

I have read that the LED should always by protected by a current-limiting resistor. ... but the point is I want to know how can the absence of R5 can cause PWM0 to go to 0V.

If you don't add R5 the PWM output will be overloaded. The voltage will max out at about the rated forward voltage, Vf, of the LED and at the maximum current that the microcontroller can give. This is neither good for the micro or the LED. Use a current limiting resistor.