This isn’t something that can be accomplished simply with Airtable’s basic functionality – it would require some fairly heavy Automation.
However, this also isn’t how Airtable is designed to be used. You will want your Table to represent a “type” of thing, in your case, that “type” of thing is a “Request”. A “Request” can have a status, but the fact that it’s status is changed to “Closed” doesn’t mean that this thing ceases to be a “Request”. Therefore, that thing should stay in its appropriate table.
Instead, what you are wanting is the distinction that Airtable makes with Views. You can create a new Grid view called “Closed Requests”, and use the Filter menu at the top to show only “Requests” where Status is "Closed".
You can have another Grid View called “Open Requests” that filters to show only “Requests” where Status is not "Closed", and this can be your main working view. When you change the status of a “Request” to “Closed”, that “Request” item will disappear from this view, so that your main working view is not cluttered with “Closed Requests”. But you can always still view and work with your “Closed Requests” by going to the appropriate View.
In this way, you are just hopping between Views, rather than hopping between Tables, and still keeping items in their appropriate place, namely the Table that represents what type of a thing it is - a Request.