Question: As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1\_R)=1\_S$. But there has been a minority who do not require this, one prominent example being Herstein in *Topics in Algebra*.

What are some of the most striking consequences of not requiring ring homomorphisms to be unital? For example, what aspects of algebraic geometry would need to be reworked if we no longer required it? What interesting theorems or techniques arise in the not-necessarily-unital theory which do not apply (or are degenerate) for unital homomorphisms?

Answer: 
Of course, if we assume that the ring have a unit, then there is absolutely no reason not to assume that homomorphisms preserve it (or are there books that do that?)

My impression is that there has been a change in the last 40 years or so; algebra texts written in the '60's or earlier mostly did not require ring to be unital, while, say, from the '70's on most of them did. Of course, rings without unit are still very much present in functional analysis (for example, Banach algebras are not assumed to be unital, as many of the standard examples are not). The first edition of Herstein's book is from 1964.

I don't know how much of this is due to Grothendieck's influence. Certainly algebraists now are less conversant with functional analysis than they used to be, due to an inevitable increase in specialization. In the kind of mathematics I practice, units are a fact of life. The only advantage I can see in not assuming the existence of a unit is that ideals become rings, and one can apply theorems about rings; but this is minor, and is more than offset by the conceptual disadvantages of working with the "wrong" category.
There are natural situations where non-unit preserving homomorphisms are needed. For instance if $R$ is a unital ring and $e$ an idempotent, then $eRe$ is a unital ring with identity $e$ and the inclusion $eRe\to R$ is not unital. Nonetheless it plays a role in Morita theory and in Chapter 6 of Green's lecture notes on polynomial representations of $Gl\_n$. It also plays a role in the construction of irreducible representations of finite monoids.