Flash an ESC Through Serial USB without Arduino Interrupting the Signal
I can offer general advice without reference to specific implementations.

By driving the Arduino pins when it is powered off you are potentially exceeding it's maximum rated specifications for voltage on any ping, and possibly for current into a pin, and you risk damaging it.
When the Arduino is powered down its Vdd pin and power supply rail will be at about 0.0 volts and will be loaded to ground by power supply capacitors and various other items connected to Vdd - including the OUTPUT of the Vdd voltage regulator in many cases.
Each Arduino pin is connected to the Vdd rail by a body / protection diode. To drive the pin high you must also raise Vdd to high-1 diode drop and must drive the Vdd rail load from your signal line via the body diode.
If your programmer has the ability to do this there is a reasonable chance that you may damage the body diode. For extra points, a very capable signal input MIGHT damage the Arduino's voltage regulator and possibly other on board devices - but I'd think that to be unlikely.

The technically best solution is to disconnect the Arduino's relevant pins when programming. As you obviously would rather not do that, then ...

Probable solution: If you power on the Arduino and run a program which ensures that the relevant pins are set to input mode then it should work as required. IF the Arduino default program sets the relevant pins to inputs then just powering it on should suffice, but odds are that the serial port pins are not both configured as inputs.
If the Arduino has a USB to serial port IC or other devices connected to the pins in question then you will need to also ensure that they do not load the pin excessively and that they will not be damaged by programming signals. eg if you were trying to drive the Arduinos serial RX_in pin which was connected to the TX_out pin of a USB-Serial IC then the programmer would be attempting to drive the USB-serial output pin and this must be dealt with in some way. 

Checklist:


Is there something else connected to this pin. 


If so 


What state is it in during ESC programming
Will it be damaged by programming
Will it affect programming.
What action is thus needed.