Is there a notion of algebraic geometry for these objects? If we take the dual category of the category of cocommutative corings with counit, is there geometry in it in a sense dual to affine schemes? Can we look at the set of coideals of a coring, put a space structure on it and sheaves (maybe cosheaves) of sections?

Let's consider coalgebras over a field rather than corings. There is a theorem that every (coassociative) coalgebra over a field is the union of its finitedimensional subcoalgebras. So the category of coalgebras over a field k is the category of indobjects in the category of finitedimensional coalgebras, while the latter is the opposite category to the category of finitedimensional algebras. Now if we restrict ourselves to the commutative case, then the category of finitedimensional commutative algebras with unit over k is the opposite category to the category of 0dimensional schemes of finite type over k, or just schemes finite over Spec k. Combining it all, the category of cocommutative coalgebras with counit over k is equivalent to the category of ind0dimensional indschemes of indfinite type over k, or just indfinite indschemes over Spec k. This explains, in particular, that the "underlying topological space" functor maps cocommutative coalgebras to discrete sets rather than anything else, and the coalgebra itself is simply the infinite direct sum of the coalgebras sitting at the points of this set. 


I like to think of algebraic geometry as being born out of the fact that Ring behaves a lot like Set^{op}. For instance, in Ring, A ∐ (B × C) = (A ∐ B) × (A ∐ C), where ∐ is the coproduct in Ring, which is just the tensor product. This formula is also true in Set after we swap ∐ and ×. This suggests that we can use our intuition about Set to think about Ring if we replace Ring by its opposite, the category of affine schemes. Cocommutative corings with counit are monoid objects in the opposite category of (Ab, ⊗). However, to get the morphisms to point in the right direction, we need to take the opposite category again: so Coring is (CAlg((Ab, ⊗)^{op}))^{op}. It follows that products in Coring are computed by tensor products in Ab, and colimits are formed by taking colimits of underlying abelian groups; and in fact Coring is a closed cartesian category, even more like Set than Ring^{op} is. In particular, we don't want to take the opposite category of Coring. Maybe this isn't surprising, since every set is already a cocommutative comonoid (w.r.t. ×) in a unique way, and we have a functor Set → Coring taking a set to the free abelian group on it. These are purely formal observations, and I don't know whether anyone has built a more concrete geometric theory, with say a functor from Coring to some kind of topological spaces with extra structure. 


One can do (relative) algebraic geometry with respect to any symmetric monoidal category, with affine schemes corresponding to the opposite category of (commutative) monoid objects. This viewpoint is developed in the paper Audessous de Spec Z (in French) by Toen and Vaquie. By taking the category of $\mathbb{Z}$modules with tensor products over $\mathbb{Z}$ one recovers usual algebraic geometry. For (cocommutative) corings, you just have to do the same in the opposite category $(\mathbb{Z}Mod)^{op}$. The big question is: what do you take as your monoidal structure? If you just dualize diagrams, your new coproduct would be the old product, i.e. direct product of the underlying abelian groups. Or instead of this you take usual tensor products. Each of these approaches will produce a different geometry. I remember a course given by Lieven Le Bruyn on KontsevichSoibelman (the noncommutative coalgebra thing) and he mentioning that by the "discrete" nature of coalgebras one could never expect them to describe anything but the etale topology of a scheme, which I think is precisely what Leonid observed in his answer. 


Leonid speaks with great authority. I'd like to point out that you get a different answer in the category of graded coalgebras. If $A$ is a graded vector space, it has a graded dual $A'$ which is smaller than its full vector space dual $A^*$. Indeed, if the grading is locally finite, then this $A'$ has the same dimension sequence as $A$ and $A'' = A$. If $A$ is a locally finite, graded coalgebra, then its ring structure is given by an infinite sequence of finite tensors, and $A'$ is equivalently a graded algebra. Graded coalgebra homomorphisms also transpose to graded algebra homomorphisms. Recall that if $A'$ is also finitely generated (and scalar in degree $0$), then it corresponds to a projective variety with a choice of an ample line bundle. The morphisms between these are a perfectly good nonfull subcategory of the projective schemes. The full dual $A^*$ is also the graded completion of $A'$. So what Leonid is saying in this case is that if you take the graded completion of a finitely generated, graded algera, it localizes the projective variety to the apex of its affine cone. This is similar to what Leonid describes for $\text{Spec}(A^*)$ in general. I said in a previous version of this answer that $\text{Spec}(A^*)$ atomizes the variety $\text{Proj}(A')$ to a "Cantorsetlike" structure. As Leonid points out in a comment, this is totally wrong in context. However, it is true that if $A$ is a finitely generated algebra, it has a completion $\hat{A}$ related to coalgebras such that $\text{Spec}(\hat{A})$ is an atomized form of $\text{Spec}(A)$. Namely, let $A'$ be the vector space of those dual vectors on $A$ that factor through finitedimensional algebra quotients of $A$. Then $A'$ is a coalgebra, and $\hat{A} = (A')^*$ is an atomization in the sense that $\text{Spec}(\hat{A})$ is a 0dimensional scheme whose points are the closed points of $\text{Spec}(A)$. (It isn't a Cantor set though.) 

