The consensus opinion regarding the sin of Nadav and Avihu, was that they entered the Holy of Holies in a state of intoxication. We know this because of the command to Aharon, that it is forbidden for the Kohein to perform Temple service after he drank wine.
There is a Gemara that advises modern day Kohanim to remain in a sober state. The Temple might fall from the sky, and sober Kohanim will be needed for immediate service.
Even today, we are careful about Kohanim drinking wine. This is why the priestly blessings are only done in the morning at Shacharit. There is no Duchaning at Mincha because Kohanim may have drunk some wine at lunch. If prayer takes the place of sacrifices, there was a morning and evening sacrifice in the Temple, and there should be a priestly blessing in the evening.
This explains why there is Duchaning at Mincha on a fast day. No wine has been consumed that day.
There is often an argument on Simchat Torah as to whether Kohanim can Duchan for Mussaf. Most synagogues give a Kiddush before Mussaf. If the Kiddush was made on wine, no Duchaning for Mussaf. I once pleaded with the rabbi of the neighborhood to allow me to give the Priestly blessing on Simchat Torah since I was careful to make Kiddush on grape juice. My request was granted.