Do airline pilots use this high-speed, high-drag technique for losing altitude? This question and another more recent one got me thinking about this.

Years ago on my MEII checkride the DPE showed me a technique for losing altitude on final when the aircraft was high. Typically, I would reduce power, increase drag (flaps, landing gear, etc), hold nose up to lose speed,   and sink like a rock (if necessary slip). However, what he showed me kind of blew my mind. I rarely do it, except when just having fun. 

He said, "Do you remember what happens when airspeed doubles?" I said, "Yes, drag quadruples." More precisely the parasite drag quadruples. Drag $= pv^2$, where $p$ is some constant representing parasite drag and $v$ is velocity. 

So, we set up high on final, and he dropped the flaps and descended at $v_{FE}$. I was surprised to find that the maneuver used up a lot of the aircraft's energy, and the aircraft was brought down to the glide path and slowed down with plenty of time for a normal landing. 

My DPE was a Captain for Southwest back when Southwest was a small airline. My question is whether this is a technique used by airline captains nowadays?

I don't have access to a B737 to try this out, but in a small aircraft like a Cessna 152 you have a $v_{s_{0}}$ of about 35 and $v_{FE}$ of 85. 

The idea is you can descend with full flaps at 42.5&nbsp;knots and power off and it will sink slowly but in a somewhat nose up attitude. Or, you can put the nose down and speed up to 85&nbsp;knots and quadruple the drag and get on your glide path sooner so that the last bit of the landing is at a proper glide path, attitude, and speed.
Airline pilots certainly do use gear/flaps/spoilers to descend more quickly if needed. However, SOP typically requires an approach to be stable upon reaching a certain altitude on the approach. This means having speed and airplane configuration set for landing, and being lined up with the runway and on the glideslope. Unstable approaches are more likely to result in landing mishaps.

Some situations such as a runway change may require late changes to the approach. It is up to the crew whether they can safely make this change or if a go-around is the better option. S-turns are another way to lose further altitude.

When Air Canada Flight 143 ran out of fuel, the pilot used his glider experience to perform a forward slip to lose altitude. These sorts of maneuvers would be reserved for an emergency on an airliner.