Many commentators attempt to turn the raven that Noah sent out into an allegory or symbol, and so, naturally, these commentators attempt to do the same thing to the dove.
Some see the raven as symbolic of an unbelieving heart, and the dove as symbolic of a believing heart (already discussed in Noah Sends Out An Unsymbolical Raven).
However, some see a different symbol. Some think that the raven and dove are symbolic of the law and the gospel. The dove represents the gospel because it brought about peace with God (represented by the olive branch it brought back to the ark). The raven represents the law because it did not bring about peace with God (represented by the fact that it did not bring anything back to the ark). The key ingredient which makes this whole allegory work is the affirmation that the olive branch symbolizes peace.
But while it is true that the olive branch is sometimes a symbol of peace, another question must be asked: When did the olive branch first become a symbol of peace? Because, if the olive branch became a symbol for peace because the dove brought back such a branch to Noah, then the allegory falls apart. But even if the olive branch was a symbol for peace before the flood, there are still some problems with the law and gospel allegory.
The raven was sent out once, and this, of course, is seen as the "sending out" of the law. The dove, however, was sent out three times. The first time it was sent out... it brought back nothing. Clearly, this creates a problem. Does this mean that the gospel failed the first time that it was "let out the window"? Quite frankly, whatever explanation is created to get around such a problem, it takes a lot of creative thinking to get this allegory to work— creative thinking which could be used for better purposes!
There are many creative thinkers who look for symbols like these, but at the same time, fully affirm the historicity of the flood— this is certainly a practice within orthodoxy. There are, however, those who deny the historicity of the flood, and in order to disguise their heresy, they attempt to impose a strange allegory upon the flood account— this is a practice that is obviously outside orthodoxy.
Noah Sends Out An Unsymbolical Dove (Part 2) - Genesis 8:8-12
Noah Sends Out An Unsymbolical Raven (Part 1) - Genesis 8:6-7
The Chronology of the Flood: The Raven and Dove - Genesis 8:6-12
The First Sight Of The Post-Flood World - Genesis 8:5
Noah's Patience - Genesis 7:24