Is there any biblical scripture that explicitly or implicitly suggests that demons have any odour?
Unfortunately there is next to nothing on this subject matter according to the Sacred Scriptures. The Bible does not explain absolutely everything, so sometimes, man has to think outside the box so to speak!
The Book of Tobit is a canonical Scripture Source in both the Orthodox and Catholic Churches, thus the vast majority of Christian denominations. There is an interesting story in relating to this subject matter in the opposite sense.
Upon arriving in Media, Raphael tells Tobias of the beautiful Sarah, whom Tobias has the right to marry because he is her cousin and closest relative. The angel instructs the young man to burn the fish's liver and heart to drive away the demon when he attacks on the wedding night. The two marry, and the fumes of the burning organs drive the demon to Upper Egypt, where Raphael follows and binds him. Sarah's father had been digging a grave to secretly bury Tobias under the assumption that he would be killed. Surprised to find his son-in-law alive and well, he orders a double-length wedding feast and has the grave secretly filled. Since the feast prevents him from leaving, Tobias sends Raphael to recover his father's money.
Here is how the Book of Tobit puts the scene:
But thou when thou shalt take her, go into the chamber, and for three days keep thyself continent from her, and give thyself to nothing else but to prayers with her.
And on that night lay the liver of the fish on the fire, and the devil shall be driven away.
But the second night thou shalt be admitted into the society of the holy Patriarchs.
And the third night thou shalt obtain a blessing that sound children may be born of you.
And when the third night is past, thou shalt take the virgin with the fear of the Lord, moved rather for love of children than for lust, that in the seed of Abraham thou mayst obtain a blessing in children. - Tobit 6:18-22
Some Catholic exorcists use this as a queue to employ ”blessed incense” in houses possessed by demons.
Strange that incorporeal beings such as demons do not like the smell of holy incense. Devils are not physical beings.
Sacred Scripture, when speaking of demons, always locates them in one of two places: in hell (that is to say, in “that which is below”) or in the air. To say that they are in the air is a way of saying that they are everywhere, that they move about with complete freedom. St. Paul repeats this again when he calls the devil “the prince of the power of the air” (Ephesians 2:2).
2 in which you used to live when you followed the ways of this world and of the ruler of the kingdom of the air, the spirit who is now at work in those who are disobedient. - Ephessians 2:2
Now that the demon are located within man’s atmosphere and being incorporeal entities, it is no wonder that Satan and his Legions are master manipulators of the atmospheric conditions that surround man.
Thus there may be no true biblical scripture source that explicitly or implicitly suggests that demons have any odour, but they know how to set the stage, so to speak, to let people know they are there. It should be noted that this is not a universal demonic phenomenon. They know all too well how mess with man’s reasoning ability.
Many exorcists (regardless of denomination) claim that they can occasionally smell the stench of evil, especially during an exorcism itself.
But the beast was captured, and with it the false prophet who had performed the signs on its behalf. With these signs he had deluded those who had received the mark of the beast and worshiped its image. The two of them were thrown alive into the fiery lake of burning sulphur. - Revelation 19:20
The demon likes to manipulate man‘s atmosphere in order to scare him into submission. He definitely does not what an exorcist and his support team to succeed, so he does what he can to enjoy his morbid freedom for a little longer.
The demon can not do true miracles, but he can manipulate the forces within the air to make it smell like hell, his real dwelling place!
Exorcisms are not for the weak hearted or the weak stomached.