Should you always band limit a signal into an opamp when you are trying to control noise - even at DC?
While I do understand the reason for limiting the bandwidth, I've seen
  a lot of schematics where there is no input filter.


The inherent noise from an op-amp, as a first approximation (e\$_{NOISE}\$), cannot be reduced by using an input filter. It can be reduced by an output filter or filtering in the feedback loop to reduce the noise bandwidth of the circuit but it cannot be removed by input filters.

Taking this to a more accurate approximation, some inherent op-amp input related noises can be reduced by input filtering such as i\$_{NOISE}\$. These for many circuits will be less than the e\$_{NOISE}\$ and are usually reduced by choosing smaller value resistors around the input stages of a circuit. This also reduces inherent thermal noise from resistors because the resistor values are much smaller.

It's sometimes good practise to limit the frequency response of an op-amp circuit to that needed for performance/spec reasons but there are many instances when this won't be the case such as when the op-amp is inherently limiting the frequency response i.e. you are pushing a design to its limits at high frequencies.