In section 3.2.5 of Nielsen and Chuang (starting page 153) they talk about Landauer’s principle, where they discuss the lower bound on the thermodynamic cost of erasing information.
In irreversible computing (i.e. classical computers), we can build any algorithm from a series of NAND gates. Since NAND gates take 2 inputs and give 1 output, a bit of information is erased (i.e. lost) and therefore Landauer’s principle is in effect.
Nielson and Chuang discuss reversible computing and how it would not incur the cost of Landauer’s principle.
Irreversible computing incurs the thermodynamic cost from Landauer’s principle and reversible computing does not. Heat and noise can ruin quantum states within quantum computers, is this related to Landauer’s principle?
I don’t see any direct statements connecting Landauer’s principle to why quantum computers must be reversible. Can someone clarify the exact role Landauer’s principle plays in the reversibility of quantum computers and its relation to the unitary nature of quantum gates?
That is not directly related. Heat and noise can ruin classical information in a classical computer too. In practice that's not such a problem because classical information, unlike quantum information, has the nice property that it can be copied. From the point of view of fundamental physics, the ones and zeros in your computer, or written down on a piece of paper, or whatever, are very redundantly encoded. In other words, there are many degrees of freedom all separately telling you "this is a one" or "this is a zero". That is not the case for a qubit. A qubit is, in a very real sense, a smallest unit of information. It occupies the smallest amount of degrees of freedom possible.
Quantum computing can be irreversible. Let me explain what I mean by that. The formalism of state vectors acted on by unitaries can be neatly extended to include probabilistic mixtures of states as well as operations that are irreversible. See Wikipedia or chapter 8 of Nielsen and Chuang. These more general operations describe the evolution of open quantum systems. Closed quantum systems (as well as closed classical systems) are of course reversible. The reversible/irreversible distinction is not a quantum/classical distinction, but rather a closed system/open system distinction.
In practice quantum computers must do irreversible operations, for example when re-initializing a qubit to $\left|0\right>$.
As for your question, I don't see why there shouldn't be a corresponding increase in entropy whenever an irreversible operation is performed on a quantum computer, just like there is when an irreversible operation is performed on a classical computer. I don't think Landauer's principle fails to hold in any meaningful way for a quantum computer.