<p>I am writing an algorithm to find the dominating set of a tournament graph. Is the minimum spanning tree of a directed graph equivalent to the dominating set of the graph? In other words, if I find the smallest MST for the tournament graph (by iterating through all of the vertices), can I then say this is equivalent to the dominating set of the graph?</p>

No, because the MST will include all vertices of the graph, and the dominating set might not.
See for example the graph here:
http://en.wikipedia.org/wiki/Tournament
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(graph
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theory)
Vertices 2 and 4 create a dominating set, and not a spanning tree.