What are the disadvantages of using a Zener diode over a linear voltage regulator?
1) When using a Zener diode as the regulation element like in this circuit: 



the disadvantage is that the circuit needs to be configured such that there's always some current flowing through the zener diode. The zener diode acts as a shunt regulator, it "burns off" the current that is "left over" instead of limiting the current that flows when little current is needed. When the load takes no current then all the current that isn't taken by the load has to go through the zener. That wastes power. In practice this circuit is only suitable for loads which draw a low current and preferably also a somewhat constant current.

Why would I use this circuit then?

Well, it's cheap.

A linear regulator like the LM7805 or a zener + transistor based circuit like this one:



form a series regulator (not shunt). These regulators have the advantage that they consume only as much current as is needed. When the load takes no current then there is only a small amount of power used.

These circuits are slightly more expensive as a transistor is needed or a voltage regulator chip like the LM7805.

2) Saying that the LM7805 is a bad regulator because it just "burns off" the excess power isn't telling the whole story. The LM7805 (and LM317 and similar) are still used a lot so they clearly have their purpose.

Fact is that for loads that do not need a lot of current, let's say up to 100 mA, then these linear regulators are a good choice.

Only when you need (a lot) more current then it might be more efficient (less power turned into heat) when a switching regulator is used. A typical example when to use a switching regulator is to convert 12 V (car or solar panel battery) to 5 V (USB) for powering gadgets. Then a current of up to 2 A might be needed. At 12 V, 2 A, a linear regulator will need to "burn off" 7 V at 2 A, that's 14 watt which requires a substantial heatsink. Even a cheap switched regulator like an LM2596 can do that much more efficiently without the large heatsink.

So don't think that some circuit solution is always better than the other. It is more complex than that. What the most optimal solution is depends on what is needed. Like input voltage, current into load, cost etc. In the real world engineers use all of the solutions I have shown here, they choose the one that best suits a certain situation.