My paper has been rejected (with a "Reject and Resubmit" status), and the Editor has recommended resubmitting it with the revision contents that incorporate the reviewer's feedback. I am considering not resubmitting it and submitting the article to another journal. Should I inform the editor that I won't be resubmitting my article to the journal?
Not necessary to inform
This is a line that the editor knowingly crossed when they sent 'reject and resubmit'. If they didn't want to increase the chance that you'll submit elsewhere, they would send 'revise'. (Functionally the two decisions are the same, but the first is more final in the sense that it increases the chances that the author never resubmits it.)
If you do resubmit it, then you should alert them that the manuscript is resubmitted (there'll almost surely be an option to do this in the editorial management system). If you don't, they'll forget about your manuscript.
Should I inform the editor that I won't be resubmitting my article to the journal?
Essentially, your paper has been rejected (Fomite, 2015). There is no obligation to formally withdraw or inform the Editor. However, there's also nothing stopping you from informing the Editor if you so wish.
I understand that 'reject and resubmit' is not explicitly covered in the Academia typical workflow of a journal.
Depending on which lens on looks at it, reject and resubmit is synonymous with revise and resubmit or major revision. Yet, there're (subtle) differences between them. They also have different implications in terms of publication metrics.