There are two notions of Zeno topics related to quantum computation. The first, which is controversial is usually called hypercomputation, which deals with the possibility of surpassing the limitations of the Church-Turing thesis by means of quantum computation. It is related to the Zeno effect through the fact that if it could be realized, it may solve the halting problem. Contogo refers to this option as a "Quantum Zeno machine". In this context, please see also:Nielsen exploring this possibility.
The second topic which goes by the name Quantum Zeno effect (as referred to in the Wikipedia page in the question) is well established and experimentally verified, (please see Kwiat, White, Mitchell, Nairz, Weihs, Weinfurter, and Zeilinger ).
Kwiat et al. were motivated by one of the most striking examples of this effect: the Elitzur-Vaidman bomb testing problem, (please see the following review by Vaidman).
This problem deals with bombs which can only interact with the outside world by means of their trigger, classically, testing the bomb would cause every good bomb to explode, but quantum mechanically, one can reach, using the Zeno effect, almost 100% detection probability without exploding the bomb. This is an example of a non-demolition measurement which is not accompanied by state reduction.
Translated to quantum computation terminology, this effect is called by Jozsa counterfactual quantum computation, according to which a quantum computer, programmed to solve a problem, can give the result even without running.
A detailed account of the Elitzur-Vaidman bomb testing problem is given by Penrose in his popular book: Shadows of the mind.
One of the most important application of the quantum Zeno effect is its exploitation to keep a system inside a decoherence free subspace (by performing repeated measurements) as proposed by: Beige, Braun, Tregenna, and Knigh. Very recently, this proposal was adapted to holonomic quantum computation by Mousolou and Sjöqvist .