can gapped systems have gravitational anomalies? The question is in the title.
If it is possible, what are some examples of gapped systems--either quantum field theories or condensed matter systems--which exhibit some kind of anomaly when coupled to a metric with curvature or placed on a spacetime with non-trivial topology?
 A: When one encounters the statement "gapped systems have gravitational anomalies", I think the precise statement is that 

"gapped systems with bulk topological order (read the link Review) have boundary gravitational anomalies".

So the answer to your question is yes, if we look into the surface of the bulk topological order.
Three papers you may look at are that:


*

*Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders

*Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

*A field theory representation of pure gauge and mixed gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
Some examples given above mentioned that gravitational anomalies exist on the boundary of bulk topological order in any dimension.
The first question you will ask is that "isn't that Alvarez-Gaumé and Witten said Gravitational anomalies
 only exists in (4n+2)D spacetime dimension?"
My best response is that Alvarez-Gaumé-Witten talked about perturbative gravitational anomalies. 
If one consider non-perturbative or global gravitational anomalies:
The topological order implies (1) nontrivial SL$(N,\mathbb{Z})$ representation through modular SL$(N,\mathbb{Z})$ transformation. (example in 2+1D SL$(2,\mathbb{Z})$ and 3+1D SL$(3,\mathbb{Z})$ with Ref here on  Modular SL(3,$\mathbb{Z}$) Representation and 3+1D Twisted Gauge Theory and here) and here; this either implies
(2) the spatial-topology-dependent robust ground state degeneracy(GSD) (such GSD depends on the genus of Riemann surface or the Betti number of the manifold, or degeneracy with gapped boundaries here and here) or (3) chiral edge modes.
Intuitively, the topology-dependent robust GSD indicates the existence of global gravitational anomalies, but more precisely how it coincides with the HEP work of Witten, we may have to dig into this further. But the basic observation is heuristic and simple.
A: The answer to my question is Yes.
Embarrassingly, one of the simplest examples is given by the fermionic quasistring topological order I described in my paper http://arxiv.org/abs/1404.4385 . The magic is that the 5th oriented bordism group is generated by the mapping torus of complex conjugation on CP^2. Thus, if we consider the fermionic quasistring top order on CP^2, the action changes by a sign when we perform the large diffeomorphism of complex conjugation on CP^2.
