The F1 visa holders get SSN. When they get GC, do they need to get new SSN Card?

No. That follows throughout your life

You don’t necessarily need to get a new one but If your SS card says “work allowed under DHS Authorization only” and you don’t like seeing that, you can apply for the replacement. The number remains the same but the card will not have that statement.

you need to update your info in ss office. Otherwise when they do e verify you will fail and have to go through hassle. I personally had that issue few years ago.

The reason you need to update in SSN office is from there it gets automatically updated to all other government agencies so you are updated in all the system.

They won't change your SSN No. They will replace your SSN card with the new one.I would recommend you to replace it.Just take your GC and old ssn card to the Social security office and get replaced. Its really easy.I replaced mine when I got my citizenship.