From the Twitter app - select the "v" icon at top-right of Tweets, select "Share Tweets via..." then select this "Wayback Machine" app.
After you have archived your web pages or Tweets you can confidently share their Wayback Machine URLs via Twitter, Facebook, Email, etc.
From Safari - select the "Share" icon at the bottom of the screen then select this "Wayback Machine" app.
But first you must add this app to your iOS device's "Share" feature by selecting the Share icon (the up-pointing arrow at the bottom of Safari) then selecting the "..." (More) button and finally adding this app.
To archive Web pages or Tweets simply "Share" them to this app.
knowing they will be there for decades to come.