Sniffing twisted pair without interupt
This is possible in theory, but will be difficult in practise.  One problem is that anything you attach to the cable will cause a impedance discontinuity, which can cause problems with the regular communication.  Depending on the situation, it also allows the right kind of sniffer to figure out you're in there eavesdropping.

You want to put a relatively high resistance immediately connected to the cable.  This resistance needs to be significantly higher than the impedance of the cable.  For example, if you're breaking into normal ethernet, then the cable impedance is 50 &Omega;.  You'd want to put at least 500 &Omega; resistors immediately on the lines so that your tap is contributing little in terms of impedance discontinuity.

However, the higher the frequency of the signals, the harder it will be to still make sense of the signals at the other end of the resistors.  This is where you may have to carefully create a circuit, physically close to the resistors, to receive the signal.  Once you interpret the signal into digital, you can then transport it by ordinary means to whatever you have decoding the data stream.