Remove DC bias without phase shift?
Are there any circuits than can remove DC bias of an input signal without inducing a phase shift?

Real life linear circuits, no. All real-world filters must be causal, that is output response occurs after the input signal, and that means phase shift. Even linear 'DC servoes' are just ways of building highpass filters with a very long time constant. The longer the time constant, the less the phase shift at operating frequency, but it's still finite. Even if you are observing a meter, and twiddling a potentiometer in response to zero out the output offset, you are part of the DC servo, and form part of the highpass filter. You can achieve the same phase shift by choosing the same time constant, whether using a conventional highpass filter, or a DC servo. There is a tradeoff between phase shift and settling time. If you choose a long time constant, then you'll have low phase shift, but a long settling time.
If we lose the linear constraint, then this is where a non-linear DC servo, or a human-in-the-loop, can improve that tradeoff. A human, observing the meter, could decide when the DC compensation was wrong, make a rapid adjustment, then leave the adjustments alone, after which there is no phase shift. That sort of behaviour is easy when you're a human, more or less impossible in an analogue circuit, and surprisingly tough to code into a processor-controlled loop so that it does what you want. Of course if it then drifts, you have to make another adjustment, and that upsets your signal.
If we lose the real-life or real-time constraint, and do offline processing of signals, then you have the ability to make filters acausal, ones which have a response before the input, which now permits you to get zero phase shift. The two easiest ways to implement them are to use a symmetrical FIR filter, or to run the same arbitrary filter both forwards and backwards over the time series. Both these filter routes guarantee zero phase shift.