Is RTS/CTS needed for UART with FT232R USB-UART chip?
This is unanswerable based on the hardware alone.

Rather it depends on the software on each end.  If the software expects (or configures operating system serial drivers or buffered UART modes to expect) the use of hardware flow control, then you must either connect these or wire them into a permissive state.

If the software doesn't try to use them (or configure for their use), then it doesn't matter if you connect them.

(In rare legacy cases, the "software" in question could be a state machine implemented in a more hardware-like fashion, but apart from UART-level buffer management this is extremely unusual - and regardless, the concept is the same, that it is the usage of the port that possibly cares about hardware flow control, not the low level of the port itself).