What type of ADC input filter for extremely noisy DC signal
You need to flip the question round the other way. It's quite easy to make a filter to a specification, once you have a spec. What you need is to derive the right spec for your application.

The whole purpose of your system is to maintain the height of your plasma cutting torch, that is, servo its voltage which is a proxy for height. I'm guessing that there exists some sort of motor that can raise and lower the torch. 

So what you have is a closed-loop control problem. I think you understand this, I notice your comment about phase lag. I am guessing that you understand this can make the loop harder to design and stabilise. But it is one thing to be nervous about phase lag when trying to design a filter, and quite another to go from the requirements of loop control bandwidth and stability, to arrive at a specification for the filter. 

One of the first things to recognise in this control problem is that the noise on the voltage measurement will get through to the torch as a height jitter. However, by designing your loop, you will be able to attenuate this jitter. In a well designed loop, it will be attenuated so much as to be negligible. 'As tight as possible' is not really a specification!

So the first major spec point is, what is the permitted height jitter of the torch at any given jitter frequency? If the frequency dependance of that spec makes it too intractable to arrive at, the total peak to peak jitter in a bandwidth can substitute, but that would need other approximations.

Knowing that permitted torch jitter, and the noise on your voltage measurement, will allow you to specify your closed loop transfer function that will achieve the attenuation. This comprises the system gains, motor lags, acceleration sensitivities, as well as any explicit filters you add.

If this is enough to put you on the control loop path, all well and good. However, it may be completely new to you, and if so, you're not going to become an expert in it overnight. This is probably too complex a situation to learn on.

So as a second best, let's turn the question back to what you asked. I guess once you have a fast enough filter, you can use it to drive a packaged height controller, and hope'n'poke some sort of stable loop.

The fastest filter, the lowest phase lag, will be to use a digital 'box-car' average function, in processing after the ADC. It simply adds all the samples together over the most recent period of time. If you are sampling at 1000Hz, then a 20mS second box-car will add the previous 20 readings together. This filter will have a latency, an average time delay, of 10mS seconds, half the length of the box-car. That time delay will set a limit on the loop bandwidth you can acheive in your loop, it may be acceptable, it may not. It is very easy to adjust.

Where your noise is expected to be periodic, and here I would expect the mains frequency to have a big contribution to the measured noise, the box-car filter will excel, by putting deep attenuation notches at the line frequency. For instance, in 50Hz land where I live, a box-car that is 20mS long (or a multiple of 20mS) will completely remove all mains frequency noise and its harmonics. In 60Hz land, make the box-car a multiple of 16.7mS long. 

The ADC will need to be preceded by an analogue anti-alias filter. The bandwidth of this filter is chosen to be just narrow enough for the sampling rate, and no narrower. This means its phase delay will be negligible compared to the digital filter and the mechanical components in the height adjust.