Hehe, bad idea - you couldn't have a beer on brewday since you'd have to stay sober to handle this with appropriate care!
But seriously, I'm wondering that if you have to ask the question about suitability then you are probably not familiar with handling liquid nitrogen. As well as getting suitable training, you would need equipment that is designed to work at those temperatures - while a copper heat exchange coil might be fine, gaskets and hoses would have to be swapped out for types that can tolerate the cold temperature and the pressure. The liquid to gas expansion at room temperature is nearly 700-fold, so great pressure can be created with risk of explosion.
Also, nitrogen isn't so great a coolant. Wikipedia says this:
Despite its reputation, liquid nitrogen's efficiency as a coolant is
limited by the fact that it boils immediately on contact with a warmer
object, enveloping the object in insulating nitrogen gas. This effect,
known as the Leidenfrost effect, applies to any liquid in contact with
an object significantly hotter than its boiling point. More rapid
cooling may be obtained by plunging an object into a slush of liquid
and solid nitrogen rather than liquid nitrogen alone.
In other words, heat exchange is effective while the nitrogen is liquid. Nitrogen is liquid between 63K and 77K - so that's at most a change in temperature of 14C (25F) before it becomes a gas and then becomes much less effective at heat exchange. So, even though the nitrogen is very cold, little of that temperature difference is exchanged with the wort. You'd need a lot of nitrogen, and it is certainly less efficient than using water.
This, combined with all the above seems to make it not worth the hassle or risk.
Looking at your original fear of the amount of water used, I use 18-22 gallons of water to chill 10 gallons of wort to pitching temps via a Dudadiesel 50 plate chiller. The amount varies with the time of year.
The output from the chiller goes back into my HLT. The water comes out warm - around 50-60C/125-140F, which I typically use some or all afterwards, so it's not wasted:
- Use some of it to clean out the boil kettle and MLT after brewday, rinsing out the pumps, chiller, HERMS coil etc..
- Mix some with oxyclean/PBW for a more intensive clean of the kettles, cleaning buckets/carboys etc..
- leave it in the HLT with the lid on and brew again the next day.
The last one is the most efficient use - by using the water again for the next day's brew, your water wastage can become much less per brew. Also heating is reduced in half since most of the heat is retained, making it also more energy efficient.