I faced a similar problem, after adding the tailwind classes to the html elements and then saved the file while the live server is running, the browser doesn't reflect the changes until I manually reload the browser.

You might be using one of the two popular live server extension, either Live Preview by Microsoft or Live Server by Ritwick Dey

To fix this issue for both of them -- Go in the extension settings and configure it as shown below

For Live Preview by Microsoft --

Live Preview --> Settings --> Auto Refresh Preview --> change it to "On changes to saved files" --> Restart the live server and you are good to go

For Live Server by Ritwick Dey --

Live Server --> Settings --> Full Reload --> Enable it --> Restart the live server and it will start working