Are there bands in EQ that can be safely quashed that are outside of human voice. Thinking that if noise is not human noise it must be room/background noise.
You could definitely lower the volume of all frequencies outside human vocal range, and this will help remove low and high frequency noise.
This will do nothing for noise within the human vocal range though, so you need to be aware that while this sort of filtering will help reduce noise, it is limited.
If you do have a specific frequency noise source within the human range, you could drop that frequency using a tight Q filter - this will affect and vocal elements on that frequency as well.
Note that due to formants, the harmonics of the human voice are much more complex than just the fundamentals + consonant sounds. It is true that some amount of highpass filtering can get rid of rumble. Another strategy for reducing perceived noise is to use a side-chain gate that opens when it detects signals that sound human. When the gate opens, the noise will be present, but it will likely be masked by all the other sounds of the human speech. When the gate closes, the noise is attenuated, preventing one from fixating too much on what a terrible job the audio team did in the first place by not having appropriate microphones, not placing them appropriately, and/or not recording in a properly treated acoustic environment.
You can use an equalizer to lower the noise "around" some voice such as dialog, but only to a certain extent.
In some cases equalizing or equalizing too much "around" the vocal leads to an unnatural tonal balance and unnatural voice (this happens e.g. if one removes "room noise" that one thinks is useless, when in fact it's what makes the recording sound natural), even if the other sounds sounded like "noise". That is, even if equalizing made the voice more clear, then overall recording may become unnatural sounding and then you've got another problem.
If it's a vocal that's going into a mix (i.e. there will be other sounds mixed with it), then the noise can be buried by the other sounds that are mixed into it and then the equalization may be redundant or behave differently.