Q: Is there any relationship between the pigeon and dove burnt offerings of Leviticus and the Holy Spirit dove symbolism of the Gospels?

A: The reason that doves and pigeons were selected is because they were quite numerous in the Land of Israel (similar to New York City in that regard) and, therefore, they were quite inexpensive. In fact, as Leviticus 5:7,11 states, only the poor who could not afford anything better were allowed to sacrifice these two kinds of birds. And Joseph and Mary were examples of this after Yeshua was born (Luke 2:22-24).

I would not try to connect these with the dove symbol of the Holy Spirit, as I do not believe this was God's intent. The primary reason the Spirit appears as a dove in the New Testament is understood through the Jewish perspective: the rabbinic interpretation of Genesis 1:2 was that the Spirit of God moved upon the face of the waters “like a dove.”

With the Jewish mindset already associating the Holy Spirit with a dove, it was quite natural for God to use that motif in the revelation of the Holy Spirit. The spiritual Jewish thought of that day would immediately draw a connection between the Spirit descending upon the Son (as the Gospels describe) and the anointing of the Holy Spirit, as depicted in Isaiah 61:1-3.