What is the Biblical basis for the concept that demons have an odor? I've seen that in Pentecostal and Charismatic churches while doing (what I would call a type of) exorcism some claim to smell the demon. They express the smell as "sour" or something rotten.

Bible Knowledge describes the smell as putrid and toxic and also like burning sulfur.

Oral Roberts claimed he could smell them in a person, and Tom Brown Ministies says it's type discernment.

According to Let Us Reason, Joyce Meyer described the smell as a "stench."

Is there any biblical scripture that explicitly or implicitly suggests that demons have any odor?
It appears that there is no explicit reference in the bible to point to anything that suggests that demons have any odor.
However, I really wanted to make sense of this (strange for me) phenomenon, so I put some thought into how to understand this.
How do you detect if something has an odor?
You smell it... stating the obvious.
How do you smell something (like coffee)?
With your nose... again stating the obvious
So, how do you smell things in the spiritual realm like demons?
With a spiritual nose or some equivalent.
Spiritual senses
The only spiritual sense that is dominantly shown in the bible is spiritual sight.
2 Kings 6:14-17 (KJV)

Therefore sent he thither horses, and chariots, and a great host: and they came by night, and compassed the city about.
And when the servant of the man of God was risen early, and gone forth, behold, an host compassed the city both with horses and chariots. And his servant said unto him, Alas, my master! how shall we do?
And he answered, Fear not: for they that be with us are more than they that be with them.
And Elisha prayed, and said, Lord, I pray thee, open his eyes, that he may see. And the Lord opened the eyes of the young man; and he saw: and, behold, the mountain was full of horses and chariots of fire round about Elisha.

Taking 2 Corinthians 4:18 (KJV) literally:

While we look not at the things which are seen, but at the things which are not seen: for the things which are seen are temporal; but the things which are not seen are eternal.

It appears from these passages that the ability to see spiritual beings is supported explicitly (for those whom God has given the ability). We can call this spiritual sight.
From spiritual eyes (sight) I would guess that it's not such a far reach to claim that other spiritual senses can exist like

spiritual ears (hear)
spiritual nose (smell)
spiritual mouth (taste)

This makes user19940's answer more complete; if something is &quot;unclean&quot; you can typically smell it (the way I check whether my clothes are clean :P); and if you can smell the spiritual realm then you could be able to smell spiritual beings like unclean spirits like demons (again if God gives you the ability).
This is a bit of a (weird) stretch answer I know, but I hope you found it at least as an interesting view point.