There is no overt reference to the seven deadly sins in this story. The seven rooms (apartments) that Poe describes are symbolic of the seven stages of life. Note that the first room is in the east, and the last room is in the west. The stages of life follow the rising and setting of the sun. There are seven days in a week. The progression suggests the passage of time: from birth to death.
Color symbolism is quite diverse. So, it might be too specific to assign a meaning to each room. The point is that the seven rooms represent seven stages in life. The first is blue, a color with many symbolic meanings, as diverse as tranquility and depression. Researching the possible significance of colors would be an interesting exercise, but we'll stick to an application of the seven deadly sins.
If Poe is saying something about human nature, let's focus on Prospero. He gathers one thousand of his upper class friends to protect them from the red death that is plaguing everyone beyond his castle walls. He believes he can cheat death with his strategy. This shows his pride. He fills his castle with plenty of food, lavish decorations, musicians, dancers, etc. This shows his greed. He also includes "Beauty" which could indicate sexual partners or maybe even prostitutes for himself and his friends. This symbolizes lust. The sin of gluttony goes along with greed and sloth. Here we have a group of privileged, upper class people who could care less about the lower classes suffering beyond their walls. This speaks to their laziness in dealing with the problem (sloth). Their greed and gluttony is illustrated by their lavish parties while people outside the castle walls are enduing death and disease. Prospero shows his wrath when he confronts and challenges the personified Red Death. Poe writes, ". . . the Prince Prospero, maddening with rage and the shame of his own momentary cowardice, rushed hurriedly through the six chambers, while none followed him on account of a deadly terror that had seized upon all." It is hard to say if envy plays a role here. One could suppose that Prospero envies the masked figure (mummer - actor) for taking the attention away from himself, but that might be a stretch.
The "seven" in this dark story does cause readers to think of the seven deadly sins. But there is not intentional connection there. However, you can apply the seven deadly sins to the actions of Prospero and his guests. Their deaths are perhaps punishment for protecting themselves and abandoning others. If Poe is making some broad statement about human nature, it might simply be that when faced with death, some people will abandon their values and embrace sin in order to save their own lives.