Are natural disasters the result of evil being allowed into our world? I remember reading somewhere that the reason there are natural disasters on this earth is because of the evil which was allowed into it when Adam and Eve sinned.  I have not been able to find this reference, though.

Has anyone else seen this, and are there any biblical references saying that natural disasters are caused by evil?
I see ALL destruction as God breathed.
All destruction and decay is because sin entered the world, before sin, this world was perfect. So how did all of this decay, destruction, evil get into the world and where does it come from?
Well, to some extent it is true that without God these things will happen, the problem you have with that theology, is that EVERYTHING was/is created by God and EVERYTHING is sustained by Him. Literally nothing that is, exists outside of God.

Colossians 1:17 (NIV)
17 He is before all things, and in him all things hold together.

God is set against the wicked every day. He is constantly set against His enemies, He promises to never give them rest.

Romans 1:18  (NKJV)
18 For the wrath of God is revealed from heaven against all
ungodliness and unrighteousness of men, who suppress the truth in
unrighteousness,

Israel's History
Look back at Israel and you'll see that every time the people were disobedient God sent destruction upon them in many fashions: plagues, famine, disease, war, earthly disaster and even miracles of disaster.
Man is the source of unjust Evil
In the last section I mentioned evil for two purposes, one for just purposes, which is the evil from God and the other for unjust purposes, which is not from God. Men on the other hand are continuously creating evil for the sake of being evil. This is what the gospel is all about. We are a wicked and filthy people, literally enemies of God.
Luke 6:45 NIV

A good man brings good things out of the good stored up in his heart,
and an evil man brings evil things out of the evil stored up in his
heart. For the mouth speaks what the heart is full of.

Matthew 13:41 NIV

The Son of Man will send out his angels, and they will weed out of his
kingdom everything that causes sin and all who do evil.

God is the source of Wrath/Evil for a just cause
Let's not discount God's ability to bring wrath and destruction onto/into the world. See in these next verses where God has actually promised to bring evil/destruction upon sinners.
Isaiah 45:7 KJV

&quot;I make peace and create evil. I the Lord do all these things.&quot;

Amos 3:6 KJV

Shall a trumpet be blown in the city, and the people not be afraid?
shall there be evil in a city, and the LORD hath not done it?

Joshua 23:15 NIV

But just as all the good things the LORD your God has promised you
have come to you, so he will bring on you all the evil things he has
threatened, until the LORD your God has destroyed you from this good
land he has given you.

1 Samuel 16:15 NIV

Saul’s attendants said to him, “See, an evil spirit from God is
tormenting you.

Understand that God does these evil acts as acts of Justice. He tells us that all of His ways are Just and He does these things because of the wickedness of the people.
Job 34:10

“So listen to me, you men of understanding. Far be it from God to do
evil, from the Almighty to do wrong.

In Job it is referring to God doing evil for the sake of doing evil and not for a just purpose.
Disasters are from God as a form of His justice.