Do respiration take place all the time in living things?

If you believe that respiration is one of the characteristics of living things , the question is self answering.

Some things that may not be considered living and which also lack respiration are :

All virus

Some dormant eggs which have survived in anoxic environments for up to 300 y.

some "living" things do not depend on an Oxygen based chemical system, but on one which depends on Sulphur. Deep under the Mediterranean Sea, small animals have been discovered that live their entire lives without oxygen . Check out deep sea vents.