There are three elements to consider when publishing certificates online:
(1) Copyright, i.e. do you have the right to publish the certificate. This will depend on how you acquired it and the constraints imposed by the provider of the certificate, and will vary from country to country.
(2) Privacy, i.e. what does the certificate reveal about living people. I would process the image to redact any information about living people before publishing it.
(3) Sensitivity, i.e. understanding the concerns of living individuals about publishing facts that the certificate reveals, e.g. suicide as a cause of death, illegitimacy or other details that some relatives might feel were best kept private. Certificates are records of facts that are often public and discoverable by other means, but each researcher must make their own assessment of how much weight they give to the sensitivities of others.
The above applies whether you publish an image of the certificate or an abstract or transcription of the information (although copyright concerns are different for transcriptions, as facts cannot be copyrighted).