We study the Minimum Sum Vertex Cover problem, which asks for an ordering of vertices in a graph that minimizes the total cover time of edges. In particular, n vertices of the graph are visited according to an ordering, and for each edge this induces the first time it is covered. The goal of the problem is to find the ordering that minimizes the sum of the cover times over all edges in the graph. In this work, we give the first explicit hardness of approximation result for Minimum Sum Vertex Cover. In particular, assuming the Unique Games Conjecture, we show that the Minimum Sum Vertex Cover problem cannot be approximated within 1.0748. The best approximation ratio for Minimum Sum Vertex Cover as of now is 16/9, due to a recent work by Bansal, Batra, Farhadi, and Tetali. We also study the Minimum Sum Vertex Cover problem on regular graphs. In particular, we show that in this case the problem is hard to approximate within 1.0157. We also revisit an approximation algorithm for regular graphs outlined in the work of Feige, Lovász, and Tetali to show that Minimum Sum Vertex Cover can be approximated within 1.225 on regular graphs.