Actually, classical physics is also reversible, whether you're considering classical dynamics or considering it as the limit of quantum physics (which is reversible).
The means for neither classical nor quantum computers, there is no operation which, in isolation, deletes a bit (that is, takes a bit in state $0$ or $1$ and sends it to $0$ in either case). Instead, you need some environmental system, and you modify the environmental system reversibly.
For example, suppose your 0s and 1s are stored as different voltage levels. If you map $0\rightarrow 0$, nothing needs to happen; if you map $1\rightarrow 0$, then the extra voltage (i.e., electrical energy) needs to go somewhere. In practice that "somewhere" is the environment, as heat.
All of this implies that the environment basically "learns" the bits of your memory when you try to delete them. This fact has two consequences: (1) Landauer's law, and (2) quantum circuits (generally) need to be reversible.
The reason this implies Landauer's law is that if you're uncertain about your computer's memory (is it a $1$ or a $0$ at the address you're deleting?) then that uncertainty gets pushed to the environment: after deleting, you know the computer has a $0$ but the environment either gained a bit of heat or not. Any increase in uncertainty about the environment translates to an increase in entropy, which translates to heat.
Since that's a complicated way to think about things, rather than model classical operations as reversible operations on the computer+environment, we model irreversible operations on just the computer and lump all environmental interactions as "heat".
(this is described in a slightly unusually way because I firmly believe probabilities are subjective, and thus so is entropy; consult other sources if you want an explanation in a different (and therefore wrong :P) perspective.)
The reason this forces quantum circuits to be reversible is that if the environment learns something about a quantum state, it "collapses" the quantum state (or in better interpretations, the environment becomes entangled with the computational state). This changes the quantum state and ruins your computation, therefore, you cannot let this happen. You need to ensure that the environment learns nothing about your quantum state, which means the quantum computer must genuinely stay isolated, and thus the operations you perform on it are reversible.
Why doesn't classical data worry about this? You can't collapse classical data!
As John Gardiner points out, many quantum computations do contain irreversible operations. In an error-correcting code, we are supposed to measure error syndromes all the time. Error-correcting codes are carefully designed so that this measurement tells us only about the errors that happened, and nothing about the computational state itself. Here, as in classical computations, Landauer's law would apply.