I have task 1 and 2 and I want task 2 to start only after task 1 is completed. If the date of task 1 gets shifted will task 2 start date will automatically changed?
No.
A link is a simple record of connection between two issues. It does not do anything other than show the human that there is some connection.
You can, of course, use the links for all sorts of things, but there is no off-the-shelf function for "update issue at the other end of a link". You will need to do some coding or find an app that can do it.
If you were to choose Scriptrunner for this, you might find some of the scripts at https://library.adaptavist.com/search?page=1&products=jira&term=LINK are a good starting point for you.
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