Common mode range and DC common mode interference
In other words is there a phenomenon called DC common mode
  interference in practice?


Yes and it's numerically embodied for any decent op-amp under the heading CMRR (common-mode rejection ratio). A good op-amp will usually have a graph indicating something like this: -



And that good op-amp will have a number quoted like this (made-up example): -

CMRR = 95 dB, DC to 1 kHz.


In addition there will be a clause to that number that says something like: -

For input voltages between -10 volts and +10 volts. Vsup = +/- 15 volts


In other words if you raised or lowered the common voltage by 10 volts the CMRR would remain better than 95 dB. Clearly this gets worse at higher frequencies due to the inability to control parasites and the falling open-loop gain.

For a differentail amplifier made from 4 resistors and an op-amp it will be a lot worse because the resistor tolerances will cause offsets.


  can there be an interference or noise source which causes increase in
  CM voltage as DC?


Anything that transmits data and uses phantom powering of a remote device is liable to this problem.