Being unital is incompatible with being coherent in the sense of topos theory: only the trivial category is unital and coherent at once (Proposition 2.10 in Cigoli-Gray-Van der Linden: Algebraically coherent categories). Hence the category of pointed objects in a non-trivial topos is never unital. It is probably easy to find further examples along these lines.

On the other hand, most pointed categories “occurring naturally” in algebra are unital. A pointed variety of algebras is unital if and only if it is a Jonsson-Tarski variety (Theorem 1.2.15 in Borceux-Bourn: Mal’cev, protomodular, homological and semi-abelian categories). This means that its theory contains a unique constant 0 and a binary operation + satisfying 0+x = x = x+0. One simple example of a variety of algebras which is not unital is the variety of subtraction algebras (which occurs for instance in Zurab Janelidze’s work on subtractivity). A subtraction algebra is a triple (X,-,0) where X is a set, 0 is an element of X and - is a binary operation such that x-0 = x and x-x = 0.