It's inspired from real events, but the way it's stated is barely recognizable and “should be regarded with suspicion” is nonsense.
Consistency has a precise meaning in logic: a consistent theory is one where not all statements can be proved. In classical logic, this is equivalent to the absence of a contradiction, i.e. a theory is inconsistent if and only if there is a statement $A$ such that the theory proves both $A$ and its negation $\neg A$.
So what does this mean regarding the lambda calculus? Nothing. The lambda calculus is a rewriting system, not a logical theory.
It is possible to view the lambda calculus in relation to logic. Regard variables as representing a hypothesis in a proof, lambda abstractions as proofs under a certain hypothesis (represented by the variable), and application as putting together a conditional proof and proof of the hypothesis. Then the beta rule corresponds to simplifying a proof by applying modus ponens, a fundamental principle of logic.
This, however, only works if the conditional proof is combined with a proof of the right hypothesis. If you have a conditional proof that assumes $n=3$ and you also have a proof of $n=2$, you can't combine them together. If you want to make this interpretation of the lambda calculus work, you need to add a constraint that only proofs of the proper hypothesis get applied to conditional proofs. This is called a type system, and the constraint is the typing rule that says that when you pass an argument to a function, the type of the argument must match the parameter type of the function.
The Curry-Howard correspondence is a parallel between typed calculi and proof systems.
- types correspond to logical statements;
- terms correspond to proofs;
- inhabited types (i.e. types such that there is a term of that type) correspond to true statements (i.e. statements such that there is a proof of that statement);
- program evaluation (i.e. rules such as beta) correspond to transformations of proofs (which had better transform correct proofs into correct proofs).
A typed calculus that has a fixed point combinator such as $Y$ allows building a term of any type (try evaluating $Y (\lambda x.x)$), so if you take the logical interpretation through the Curry-Howard correspondence, you get an inconsistent theory. See Does the Y combinator contradict the Curry-Howard correspondence? for more details.
This is not meaningful for the pure lambda calculus, i.e. for the lambda calculus without types.
In many typed calculi, it's impossible to define a fixed point combinator. Those typed calculi are useful with respect to their logical interpretation, but not as a basis for a Turing-complete programming language. In some typed calculi, it's possible to define a fixed point combinator. Those typed calculi are useful as a basis for a Turing-complete programming language, but not with respect to their logical interpretation.
- The lambda calculus is not “inconsistent”, that concept does not apply.
- A typed lambda calculus that assigns a type to every lambda term is inconsistent. Some typed lambda calculi are like that, others make some terms untypable and are consistent.
- Typed lambda calculi are not the sole raison d'être for the lambda calculus, and even inconsistent typed lambda calculi are very useful tools — just not to prove things.