 
This is not an easy subject. It is all contingent on one's intentions. Although we would all like the Lord to simply take our precious pets in their sleep after a ripe and gentle old age, this is not always our reality.
First - Euthanasia is never to be used for people no matter how much we don't want them to suffer. This is between them and God, with God calling the shots.
With animals we have a different dynamic.
1. Animals are sinless. There is no reason for us to be concerned about how they stand with God.
2. We are allowed to take an animal's life if we need it for food. This was granted to us after the Great Flood because of the sinfulness of man. But God has also fenced this around with provisions which acknowledge that their life belongs to God alone.
3. The Old Testament is full of anti-cruelty to animals laws. Even if you hate the owner you are to have mercy on the animal.
Sin
The concept of Sin pertains to the attitude of your 
heart.
Are you granting mercy (freedom from inescapable pain 
and suffering) to a cherished pet or simply 
getting rid of a nuisence?  This attitude, the 
condition of your heart, is what determines whether 
euthanasia is a sin or not.
Things to do.
1.  Pray.  Prayer is very important.  Pray about your 
decision.  If your pet is very ill don't wait to start 
praying.  Ask God's guidence in the treatment and care 
of your sick friend.  Take advantage of what medical 
care you can.  Ask God to tell you what is right and 
what is enough.  And then believe it.  If the Lord 
pierces your heart with the knowledge that it is time 
to let go, don't doubt it, don't second guess it.  God 
does answer prayer and God does speak to His 
people.
Please remember, the goal is quality of life, not 
quantity of life.  If we go beyond what is enough then 
we are not doing right by our friend.  We are being 
selfish and inflicting suffering where we are to show 
mercy (love in action), and this is a sinful 
attitude. 
2.  If you decide to euthanize your pet be with him/her 
if 
at all possible.  Let him know he is loved.  It is good 
for him and it is good for you.  
If you can't be there, then talk to your vet.  The vet 
can help you make the best decision for all concerned.  
But remember, you are making the decisions with God to 
strengthen you.  The vet is not making the decision. 
3. Don't just leave his body. You most likely will regret it. Closure is important for you and other family members. You can cremate or bury, or both. You can read prayers and have a small funeral service if you like. (There are prayers which you might like to use on the Orthodox prayers for & with our Fellow Creatures site linked below.)
4. At night when you pray thank the Lord for blessing your life with the love he sent to you in this His small creature. And thank Him for enriching your life and supporting you through the easy and the difficult.
Orthodox Prayer for & with our Fellow Creatures
 
"Speak a word to me, Father."
 
Copyright © 1999, L.A. Christensen. All rights reserved.