Applications of shortest path problem
The classic application is analysis of graph data. For example, protein-protein interaction networks or gene regulatory networks and the like. There are various types of data that can be presented in the form of a network and network analysis is an integral part of Systems Biology. 

Another example is determining the similarity of functional annotations from the Gene Ontology. One of the ways of doing so is finding the shortest path between two nodes in the GO graph.

Or, for a more concrete example, I recently published a paper where I examined various features of a particular group of proteins. One of the pieces of information I used was how central these proteins were to the network and, for this, I measured how often they served as a node in the shortest path between two others. 

The simplest way for you to find more example is to go to PubMed and search for "shortest path":

https://www.ncbi.nlm.nih.gov/pubmed/?term=shortest+path