What is the overhead of RMAN hot backup (full backup including archive logs) being performed during business hours?
Does it actually affect the performance of the database?
What is the case when incremental backups are taken in terms of overhead?
That depends on your disk subsystem.
A full backup is going to involve a metric crud-load of I/O (that is a technical term). That is going to put a huge load on your I/O subsystem. Depending on how I/O bound your system is, how much spare bandwidth your I/O subsystem has, how fast your I/O subsystem is, etc. the impact will range from "yeah, maybe it's a bit slower" to "everything times out, the system is unusable, users will be visiting the DBA team shortly with flaming pitchforks to express their displeasure."
I am hard-pressed to imagine a whole lot of situations where I'd ever want a full backup running on a non-trivial production database during anything close to business hours. If you have a real need to do this, I'd tend to be investigating things like DataGuard that would potentially allow you to offload the load of doing the backup to the standby database.
I confirm that a full or incremental backup during business hours can make your database inaccessible to end users and web services. Factors that can make things worse: