I'm taking a computer system design course and my professor told us that in digital systems, the conventional voltages used to denote a digital 0 and a digital 1 have changed over the years.
Apparently, back in the 80s, 5V was used as a 'high' and 1V was used to denote a 'low'. Nowadays, a 'high' is 0.75V and a 'low' is around 0.23V. He added that in the near future, we may shift to a system where 0.4V denotes a high, and 0.05V, a low.
He argued that these values are getting smaller so that we can reduce our power consumption. If that's the case, why do we take the trouble to set the 'low' to any positive voltage at all? Why don't we just set it to the true 0V (neutral from the power lines, I guess) voltage?
You are getting confused. Look at TTL for example: -
A low input level is between 0 volts and some small value above 0 volts (0.8 volts for the case of TTL).
why do we take the trouble to set the 'low' to any positive voltage at all?
We take the trouble to ensure it is below a certain small value.
Picture from here.
You are confusing the "ideal" value with the valid input range.
In usual logic, in ideal conditions, the logical zero would be precisely 0V. However, nothing is perfect in real world, and an electronic output has a certain tolerance. The real output voltage depends on the quality of wires, EMI noise, current it needs to supply etc. To accommodate these imperfections, the logic inputs treat a whole range of voltage as 0 (or 1). See the picture in Andy's answer.
What your lecturer probably meant by 0.75V is one of the points making the logical 0 range.
Note there is also an empty range between 0 and 1. If the input voltage falls here, the input circuit cannot guarantee proper operation, so this area is said to be forbidden.
The mapping of logical values to voltages is arbitrary. It really depends on the technology you use. In the early 80's, TTL chips were commonly used, so 5V and 0V, defined '1' and '0'. Later on, CMOS technology adopted these levels for compatibility. In the early 90's, CMOS technology advanced enough to allow lower voltages and that progress continued to this day.
Now, consider this, Binary is only one mapping. There are multi level signaling technologies, where 0, 1, 2 and 3 are mapped to some voltage levels.
The actual limit depends on the semiconductor we use, and its band gap voltage. With silicon (and current doping), it is ~0.7V, below that, switching cannot happen.