Does one ever need infinitely many cohomologies?

I will try to say something about the string theory part of the answer.

Here there are some examples where an infinite number vector bundles/sheaves are important:

1) Holography: All known examples of quantum field theories with gravity duals have large $N$ limits. For large $N$ dualities to work, a gigantic number $N$ of color charges are needed. In the physical AdS/CFT correspondence that translates into the fact that there are allowed interactions in between an infinite number of "color bundles"(talking about the CFT side).

See Yin's talk on the computation of $1/16$ BPS states in $N=4$ SYM for an interesting example of a situation where an actual computation of an infinite number of representatives of Lie algebra cohomology classes is important Xi Yin - Comments on BPS states in N=4 SYM.

2) In topological strings and topological field theories the situation is rather similar. Large $N$ dualities in those contexts requiere an infinite amount of (quasi-coherent but not coherent) sheaves supported over a homology class. Then, to exactly compute the spectrum of open strings or gluons (and their interactions) we need to take into account an infinite number of sheaf cohomology classes at the same time. See On the Gauge Theory/Geometry Correspondence.

3) Quiver gauge theories: The possible fractional branes in a quiver are identified with the derived bounded category of quiver representations (or coherent sheaves in the associated quiver variety); see Topological branes from descent for a proof of the fact that any object in the derived category is a boundary condition for the B-model. The fact that every complex is bounded (all their nodes are zero, except for a finite number of them) does not rule out the possibility that an infinite number of complexes of sheaves could be relevant in some computation. See crystal melting and black holes for an interesting example.

4) Topological gravity: All the amplitudes in Kodaira-Spencer theory of gravity can be computed enforcing an infinite number of conditions coming from the symmetries of an infinite dimensional algebra. See Topological Strings and Integrable Hierarchies and Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions.