As a matter of fact, it does.
In module 2 there is an exercise that requires turning off the air to simulate and OOA situation and give the student an idea of what it feels like when the regulator/tank is breathed down. The objective of the skill is to give students a feeling of what it's like to run out of air in controlled circumstances and to teach responding to an OOA situation with the correct hand-signal.
The guide to teaching also suggests closing the valve during the OOA skill in module 3, although a lot of instructors do not follow this advice.
R..
Rob is correct, although I personally believe that the exercise as designed has little value.
My theory (and I don't know for sure) is that when this exercise started decades ago, the quality of regulators was not what it is today. In the pool, as in normal recreational diving, you would find it increasingly difficult to breathe as the regulator got near OOA. This exercise showed students what it was like so they would have a warning before going OOA and thus have time to begin doing something about it. Today's higher quality regulators do not give such a warning before going OOA in the pool and at shallower depths. You are just suddenly OOA. As soon as you feel yourself go OOA and signal OOA, your instructor turns on the air and you will have it instantly. Thus, you experience being OOA for about a second--maybe less. As designed, there is no benefit to that drill.
If you go OOA on a deeper dive, though, you will indeed feel the regulator getting harder to breathe. I have a very fine regulator that I use with a stage bottle (an extra tank carried to increase the volume of gas on a long dive), and on some dives I breathe that down to empty and then switch to the gas on my back. I usually make the switch before I get right down to the end, but on occasions I have waited longer than normal and felt that change. I have gone through several hard-to-breathe breaths before making the switch, so I know it really does give ample warning at depth.
Consequently, I do the PADI training exercise differently than designed. I watch the student's SPG carefully, and as it approaches zero, I crack the valve open ever so slightly. This does indeed give the same kind of hard-to-breathe experience one would feel at depth. In my case, the students are never actually OOA, but they get to feel what it is like at depth.