Originally, I misunderstood the question, and was answering a question like "Is it true that quantum computers are necessarily formulated only out of reversible gates?". However, I now understand that the question was intended to be "Must there always be a reversible step inside a quantum computation?"
No - there are some computational schemes, such as the this paper, in which every computational step can be formulated as a dissipation, which is non-reversible.
Another way to think about it is in terms of the measurement-based model. Here, one must prepare a particular state, and perform a sequence of measurements on it. Usually, we talk about the initial state preparation as a unitary, but that's not necessary. The usual cluster state is the common eigenstate of a set of stabilizers $\{K_n\}$ ($K_n^2=1$ and $[K_n,K_m]=0$). So, imagine you take some initial state of a system, and start measuring sets of qubits according to the stabilizers $\{K_n\}$. The state people normally talk about preparing is the +1 eigenstate of all of these. Obviously, it's quite unlikely the we succeed in preparing that. However, we will have a record of which stabilizers gave -1 answers. It's a particular property of these stabilizers that each one anti-commutes with a $Z$ on a particular site, while all others commute with that particular $Z$. This means that the list of stabilizers with -1 values corresponds to a list of sites with $Z$ errors. But, knowing that, we can just incorporate the existence of the $Z$ rotations into the measurement bases of the computation. Everything can be achieved through measurements! (Although you require 5-qubit measurements. While not impossible, I think most people would consider implementing them using some unitaries and single-qubit measurements, but the formalism doesn't require it.)
Original Answer
No. Usually, we phrase quantum algorithms in terms of unitaries (reversible) plus measurements (not reversible). You might even argue that for many algorithms, with a deterministic output, the final measurement does nothing, and hence does not do anything irreversible.
However, this is partially the fault of the gate model which hides measurement at the end. In fact, there is some degree of choice about trading off between the two. For example, measurement-based computation implements one very simple unitary operation that may have nothing to do with the computation to be performed, and then the computation itself is specified by the measurement choices, and the measurements also detect the outcomes. This extreme certainly shows that it is not necessary to have reversibility.
