USB_PWR is connected to rocker switch, that switches between usb power and battery power.
3V3OUT connected to get 3.3v internally generated from ic.
When am powering the circuit from the battery, txd and rxd are sending some data which is giving voltage to ft232rl thus making ds3 LED glow.

USB_PWR is connected to rocker switch, that switches between usb power and battery power.
A rocker switch, whose bright idea was that?
3V3OUT connected to get 3.3v internally generated from ic.
Great so you have a low impedance path from a 3.3 Vout output to a source of +5 Volts and you wonder why the chip is having problems.
When am powering the circuit from the battery, txd and rxd are sending some data which is giving voltage to ft232rl thus making ds3 LED glow.
Only one of them can have a signal that sources or sinks current, and I don't believe there is enough enough current coming out of a GPIO pin to damage the protection diodes in the FT232RL. The question is what state is USB_PWR in and what voltage can be measured on USB_PWR and 3V3OUT which are shorted together and all by itself is enough to cause major problems.

 

3V3OUT connected to get 3.3v internally generated from ic.
Great so you have a low impedance path from a 3.3 Vout output to a source of +5 Volts and you wonder why the chip is having problems.
When am powering the circuit from the battery, txd and rxd are sending some data which is giving voltage to ft232rl thus making ds3 LED glow.
Only one of them can cave a signal that sources or sinks current, and I don't believe there is enough enough current coming out of a GPIO pin to damage the protection diodes in the FT232RL. The question is what state is USB_PWR in and what voltage can be measured on USB_PWR and 3V3OUT which are shorted together and all by itself is enough to cause major problems.

When I measure USB_PWR it is around 2.5V that is the reason for LED to glow. USB_PWR is not connected to anything.it will power the board through USB_PWR (3.3v) when rocker switch is turned. Until I turn the rocker switch the board including microcontroller is powered by battery voltage.

 

When I measure USB_PWR it is around 2.5V that is the reason for LED to glow. USB_PWR is not connected to anything.it will power the board through USB_PWR (3.3v) when rocker switch is turned. Until I turn the rocker switch the board including microcontroller is powered by battery voltage.

I still can't see the rest of your design and your statements lack credibility. USB_PWR is connected to VCCIO and 3V3OUT. How can you sit there and tel me it is not connected to anything. Your schematic looks like it was done by somebody who never even looked at the datasheet for the FT232RL. You need to go back and review this design IMHO.

One more thing. There is no external crystal, so you must be using the internal crystal. The datasheet SAYS that VCC must be 4.0 Volts or greater to enable the onboard 3.3 Volt regulator. You have CLEARLY violated this requirement. Your own measurement and statement confirms that the FT323RL DOES NOT HAVE a proper power supply. You can hardly expect a chip without a power to do much more than sit there dumb and stupid.

I won't speculate on the health of the chip, but it could have survived this mess.

 

I still can't see the rest of your design and your statements lack credibility. USB_PWR is connected to VCCIO and 3V3OUT. How can you sit there and tel me it is not connected to anything. Your schematic looks like it was done by somebody who never even looked at the datasheet for the FT232RL. You need to go back and review this design IMHO.

Sir I mean to say ,USB_PWR is connected only to what you see in schematic I have attached and nothing else outside.
I have made this schematic from the datasheet of ft232rl. The circuit has problem when I Don't connect a USB cable and power it from battery.

 

Sir I mean to say ,USB_PWR is connected only to what you see in schematic I have attached and nothing else outside.
I have made this schematic from the datasheet of ft232rl. The circuit has problem when I Don't connect a USB cable and power it from battery.

Your schematic doesn't show where Vcc (pin 20) is connected. You still have not addressed the problem that come hell or high water you need to see 3.3 Volts on USB_PWR in all conditions (battery or USB) and probably in excess of 4.0 Volts on VCC in all conditions (battery or USB) because you are using the internal oscillator. If you did this design from the datasheet and it's not working, then you must have missed something.