The first issue is that you may not have sounds enabled from within lichess. The sounds can be enabled by pressing the speaker icon on the far top right of the page.
"I can't see the speaker icon."
If you can't see the speaker icon, it's probably because your browser does not support the sounds files lichess uses. Safari and Internet Explorer have dropped support for .ogg sounds. For all intents and purposes it is recommended that you use Google Chrome (or Chromium), or Firefox while using lichess.
"The sound icon is enabled but I can't hear anything."
Make sure that you have system sounds enabled, or your speakers are turned on. Test to see if your browser can make sounds by playing a Youtube video or something similar.
"I'm sure that there should be sounds playing. I've tested all these issues and nothing has worked."
You may have encountered a bug. When using another tab, it is not uncommon for sounds to not work. Fixes have been made to resolve this issue, however if the problem persists, please contact a member of staff.