However your biggest concern will be that logging into an account from a different area & ISP than the user will raise security alerts and lock the users out. I suggest you make a client-side solution instead (a desktop app or even a browser extension) so it operates from the same IP address that the user has.
By doing that, you’re app will be almost immediately blocked, hopefully banned, and likely also the user accounts banned.
The right way to do it is using fb api under its limitations. In addition, if I recall correctly, you’ll need to refresh user’s token every 30d, so you prob want an excuse for your users to come back to your site frequently, e.g. via an email that shows weekly stats.
Automation of tasks is in demand and is a good win-win business model.
There are different ways to do that - API could be the one, and different services do store users passwords to help users to do what they want and need.
No need to scream "banned".
GP brings up a good point about some recurring login mechanic, maybe something like that would work.
PS. Any other recommended sites to discuss this topic?