How do I archive messages on iPhone?

There is no archive function for Messages.

The most you can do, is take screen shots of your messages and save them in the Photos App.

suekephart wrote:

How do you archive messages?

I archive mine using the computer app iMazing. It allows me to save my messages as pdf or text files. TouchCopy is another, similar application.

Tormhe wrote:

you can’t achieve messages but you can unmute them

Archiving and muting are entirely separate and largely unrelated features. Usually, when people want to archive messages, it means they want to remove them from the phone and store them safely somewhere.

I archive my messages to pdfs. PDFs are largely device-independent. That means, should I ever decide I'm done with iPhones, I'll still have a record of my messages. I can even print them out.

Tormhe wrote:

you can’t achieve messages but you can unmute them

What does muting or unmuting them have to do with archiving?

suekephart wrote:

I thought this was a "smart phone"!

Unfortunately, "smart phone" is not synonymous with "phone that does absolutely every little thing I think it should."

You can let Apple know what you'd like to see here:

Product Feedback - Apple

It is a smart phone. It does a veritable ton of smart things, there are also many things it does not or cannot do, but that does not make it any less smart.

The fact it doesn't do that one thing you want it to do, does not in any way diminish its smartness.

you can’t achieve messages but you can unmute them