Typically, the user who posted a story to Instagram
is able to see who viewed it.
But here are two ways
to view an Instagram story anonymously.
Using an Instagram account
that isn't automatically identifiable as you
might be the best way to stay anonymous.
If you already have an account,
you can easily add another and switch between the two.
Tap on your profile name and then "Add account."
Tap "Create New Account."
Follow the prompts to set up an account,
and make sure your username and profile picture
doesn't include your name or face.
Signed in to your new account,
view the Instagram story,
and the person will only see the new account details.
The caveat here is that the account
you want to view the story on must be public,
or else they'll have to add your new account
as a friend to grant you access.
You can turn airplane mode on
just before viewing a story to enable anonymous viewing.
Connected to the internet,
go to the profile whose story you want to watch.
Turn on airplane mode.
Now tap to view the story.