If Arduino connected to computer and opened up Serial Monitor, will Tx and Rx pin still be working?
Suppose one connected two of the Arduino Uno boards through Rx and Tx
  pins.


I assume you will connect the TX of one Arduino to the RX of the other
one and vice versa, which is the right way to connect them. As
theSealion explained in his answer, you shouldn't connect two TX Pins
together.


  if one or both of the Arduino boards were connected to computer, and
  possibly opened up Serial Monitors


If you do that, the Arduino won't even notice.

Well, to be more accurate... there is a small chip on the Uno, near the
USB connector, labeled “MEGA16U2”. This chip is used as a USB/serial
converter, and it will notice when you connect the USB link and when
you open the port. However, the main chip (the big PDIP28), which is the
microcontroller that runs your sketch, will neither know nor care.


  will Tx and Rx pin still be working?


Absolutely.


  will Arduino still communicate through Rx and Tx pin when it's
  communicating to computer?


The RX will not communicate with the computer. What will happen is:


Whatever your Arduino writes to its serial port will be received by
both the other Arduino and the serial monitor.
Whatever you send through the serial monitor will be lost.
When you Serial.read(), you will get whatever was set by the other
Arduino.


The reason of this behavior is a 1&nbsp;kΩ resistor that sits between
the Mega16U2's TX and the main microcontroller's RX. When you connect
this RX to the other Arduino's TX, which is low impedance (≈ 25&nbsp;Ω),
that TX will “win”: a low impedance output overwhelms a high impedance
one.