This would depend on the country and the type of record.
England/Wales census images are copyright (I believe by the National Archive). Images of parish registers and civil records (wills etc.) are often copyrighted by the county archive which holds them.
As a general rule it is best to work on the assumption that images are copyrighted. It is a breach of copyright to reproduce a complete page, although a snippet is generally considered "fair use".
Philip
ADDED - Guy's response is (as ever) clear and comprehensive)
No, in general the census records for England & Wales with the exception of the 1911 census, are Crown Copyright. The 1911 census is a special case as each householder completed the forms therefore it could be argued that the form is Crown Copyright, but the details hand written on the form are the copyright of the householder or his/her heir if deceased.
However there is a complication as I believe the OP is really concerned with the images rather than the data. In that case it could be argued that the copyright of the image belongs to the company who scanned the image, due to the skill of scanning each separate image, rather than simply digitising a film with a film scanner.
Parish register are the copyright of the incumbent of the parish (if it still exists) though in some cases that may be administered by the diocese Archive which may indeed be the County Archive.
Wills are the copyright of the deceased person who wrote the will, the archive who holds the will is not the copyright holder no matter what they may claim unless it was specifically passed to them.
As for fair use that changes if the image is published online as it is not simple one use of the snippet but possibly thousands of uses of the image as each time it is viewed counts.
The problem with copyright law is it is not only complex but open to interpretation and challenge and case law has changed with different challenges.
In some cases the law changes not because of the quality of the challenge but the ability of the copyright owner being willing or able to fund the defence of the challenge.
Cheers
Guy