According to the Bible, do animals have souls?

Any Biblical support? Yes; the writer of Ecclesiastes takes it as a given (Ecclesiastes 3:21, ESV quoted):

Who knows whether the spirit of man goes upward and the spirit of the beast goes down into the earth?

Also, the book of Revelation appears to refer to creatures in the sea as living souls - some translations do say living things, but the Greek word is ψυχή "breath, soul, spirit" (Revelation 16:3, KJV quoted):

And the second angel poured out his vial upon the sea; and it became as the blood of a dead man: and every living soul died in the sea.

Either way, if animals do have souls, your author is right that they may differ in important ways from human souls.

What effect are you trying to observe when you ask whether animals have souls? Or what properties associated with the English word "soul" are you looking to find in animals? A better directed question will help quell the quibbles about definitions appearing in every other answer here.

I think the fundamental difference between the animals and humans is not the soul, but the presence of a spirit. I think that distinction can be derived from Genesis 1 (KJV):

²⁴And God said, "Let the earth bring forth living creatures according to their kinds—livestock and creeping things and beasts of the earth according to their kinds." And it was so. ²⁵And God made the beasts of the earth according to their kinds and the livestock according to their kinds, and everything that creeps on the ground according to its kind. And God saw that it was good.

²⁶Then God said, "Let us make man in our image, after our likeness. And let them have dominion over the fish of the sea and over the birds of the heavens and over the livestock and over all the earth and over every creeping thing that creeps on the earth."

²⁷So God created man in his own image, in the image of God he created him; male and female he created them.

In verses 24-25, God describes his creation of the animals...that he made the animals "according to their kind". In verses 26-27, God describes his creation of man, and clearly delineates that man was made after HIS OWN IMAGE AND LIKENESS. The likeness of God is that of an eternal being with a spirit, the Holy Spirit.

This distinction is made clearer in other passages, such as 1 Thessalonians 5:23 (KJV):

Now may the God of peace himself sanctify you completely, and may your whole spirit and soul and body be kept blameless at the coming of our Lord Jesus Christ.

Where the spirit is described separately from the soul. The spirit is also described in 1 Corinthians 2 (KJV):

¹²Now we have received, not the spirit of the world, but the Spirit who is from God, so that we may know the things freely given to us by God, ¹³ which things we also speak, not in words taught by human wisdom, but in those taught by the Spirit, combining spiritual thoughts with spiritual words. ¹⁴But a natural man does not accept the things of the Spirit of God, for they are foolishness to him; and he cannot understand them, because they are spiritually appraised. ¹⁵But he who is spiritual appraises all things, yet he himself is appraised by no one.

Which brings clarity to the purpose of a healthy, living spirit... It is the spirit that is rejuvenated or given to us when we are born again. It is our spirit that gives us the ability to commune with God via his Holy Spirit. It is the part of our being that, because it is in the likeness of God, allows us to commune with him. There are differences between different translations of the bible, and some seem to indicate that unsaved man does not have a spirit, while saved man does. I am unsure what that means, or which translations are accurate. However it is an intriguing state, one which may beg the question...are we no better than animals until we accept salvation from God? Are we simply more intelligent beings with a soul until the day we repent of our sins and ask God for forgiveness and eternal life in heaven? I am not educated in the word or its language roots enough to answer that question myself, so hopefully someone more versed than I am will come along and provide a more definitive answer.

Regarding animals, there is no reference to them having a spirit in the bible. It is intriguing to note, however, that animals are referenced at least once in Revelation (that I can think of):

Revelation 19:14 (KJV):

And the armies which were in heaven followed him upon white horses, clothed in fine linen, white and clean.

I do believe God cared for the animals, despite the difference between them and man. He gave mankind, the beings he created in his own image and likeness, his most cherished creation, dominion over the animals and the earth, so he must have cared for the animals. I cannot say for sure what the passage from revelation means, however it seems that there will be animals in heaven as well. Whether they are the resurrected/eternal souls of animals on earth, or new creations, I cannot say.