Is there a difference in how long a beer is considered good when running through a nitrogen system vs a co2 system?
I am going out on a limb and interpret "stay good" as "stay carbonated/nitrogenated." That being said, carbonation of beer has a lot to do with temperature and pressure. If you are using a kegerator, you need to pay attention to: the temperature of your beer; the distance the beer has to travel to the tap; how well insulated that line is; and how much pressure you have delivering that system. If you are trying to carbonate your home-brew with nitrogen, you will need to rack the beer into the keg and then lower the temperature before force carbonating with Nitrogen. If you're just trying to drink a nitrogenated beer, provided ideal storage conditions (little to no light, cool, undisturbed) there should be a "best by" date on the bottle somewhere.
Why Are Nitrogen Beers Are "A Thing."
Traditionally, cask-conditioned English ales were served with a hand pump, or “beer engine.” The barrels in the cellar of the pub were connected to the tap faucets on the bar upstairs, and the beer was drawn up by pumping air into the barrel. These hand pumps delivered about 25 to 30 pounds per square inch (psi).
Air is 78 percent nitrogen, so they were pumping beer that had a low level of carbonation with something that was mostly nitrogen.
The trouble with using air to pump beer is that the remaining 21 percent of air is mostly oxygen. If a keg is going to be consumed quickly, using air is okay. But if the beer is consumed over a period of more than one day, it quickly becomes oxidized. To avoid oxidation but duplicate the effects of traditional cask ale, modern pubs pump their low-carbonation draft ales with a mixture of 75 percent nitrogen and 25 percent carbon dioxide at high pressure, about 28 to 30 psi. Homebrewers can do this too, but using nitrogen is not as easy as CO2.
It’s important to understand some basics. Nitrogen doesn’t behave like CO2. Nitrogen doesn’t have the affinity for going into solution like carbon dioxide does. Professional breweries “nitrogenate” their beer by chilling the beer to 32° F and forcing nitrogen into it under extremetly high pressure. Nitrogenation helps brewers re-create the smooth feel and thick head of cask ales. If you tried brewing a batch of homebrew, moving it into a keg, then pressurizing with nitrogen for a number of days, the only result would be a tap delivering very flat, high-pressure homebrew. As a safety note, most homebrew equipment is not designed to handle the pressure needed to force nitrogen into beer.
Information for Nitrogen-Homebrews
First, make sure the beer recipes you want to dispense will be good at 55° F with low carbonation. The best candidates for this treatment are traditional stouts, porters, milds, and bitters.
Good hop flavor and aroma and the flavors from specialty grains are important in beers with low carbonation. If you have some of your favorite beer bottled, open one and let it sit for a couple of hours before drinking it. If it seems better that way, that recipe is a good candidate for a traditional English draft beer.
To be prepared for traditional serving, a beer must have a low level of carbonation. If you are adding priming sugar when kegging, cut the amount to one-third cup of corn sugar in five gallons of beer. If you are artificially carbonating, put the keg in the refrigerator, apply 20 psi CO2 pressure and give it a shake. The next day, take the keg out of the refrigerator, let it warm up to 55° F, and draw off some beer to check; there should be some carbonation, but not much. Stir the beer with a spoon and see if some CO2 bubbles are knocked out of solution.
When you feel that the beer has the proper low level of carbonation, keep it at 55° F and reduce the CO2 pressure to 5 psi to maintain it. A stick-on strip thermometer on the keg is very helpful for checking the liquid temperature.
I don't see CO2 versus nitrogen making a difference in freshness. You are probably thinking that because nitrogen is an inert (ie non-reactive) gas that is used in some packaging to extend the shelf life of some foods like fresh pasta, using it instead of CO2 would extend the life of your beer. However, beer already has significant amounts of CO2 dissolved in it so using nitrogen isn't going to keep it any fresher as a result.
Nitrogen is used versus CO2 because some people like the texture it creates as opposed to CO2, however that's a separate topic which has already been covered in another question.
TL:DR; Not a lot of difference, but science favors carbonation for long-term storage. In a keg system, long-term storage isn't a factor, and there's no appreciable difference.
In order to answer your question, a couple of things, first. 'Stay good'; is somewhat nebulous, but let's say that it consists of three things; 1. Not being oxidized or lightstruck 2. Not being infected 3. Being pleasant to taste
Point 1: Oxidation can be minimized by using good brewing practices, which (if you're bartending) is kinda out of your control. If you're the brewer, that's different. But oxidative compounds can be in your hops or grain, so that may still be out of your control. Lightstruck is easy to avoid with good storage practices and brown bottles [- first google listing on beer cellaring] here, and it has good info.1.
Point 2: Beer isn't sterile. If it were sterile, it wouldn't be good. Pasteurization can make beer nearly sterile - so close that it pretty much is sterile - but pasteurized beer generally has a shorter shelf life than beer with the right active microbes. Case in point: a pasteurized pale american lager at 3 years vs. a non-pasteurized lambic at 3 years. Or a bottle of Brett-aged beer. Lambics and wild beers have tons of microbes. Microbes do good things for stability, long-term. Or bad things, if you have bad microbes.
The reason they stay good and don't make you sick is that the yeast, lactobacilli, pediococcus et al, alter the Ph so that hostile microbes won't grow, which leads me to the finale of point 2:
Co2 in solution drives the Ph of the liquid down, or towards acidity. Hard science paper about that here(although you can only read the preview without membership, there's a graph in the preview that shows the effect on atmospheres of dissolved Co2 vs Ph).
Point 3:Being pleasant to taste. Although this could be considered subjective if we got into the flavor of N vs Co2, it really just hearks to point 1: no bad taste chemicals. Good storage and good brewing make for good beer.
I looked all over for examples of nitrogenated beer that would be considered cellar-able. I didn't find any. Does this mean that Co2 is better for the health of your beer?
Not necessarily. In order for N to come out of solution, it needs nucleation sites, which are usually other N bubbles. Coming out of a tap, it has bubbles. With a widget, it also has bubbles, but I'd be concerned about the alcohol slowly degrading a widget, were I to store nitrogenated beer long-term. Plenty of science out there about degradation of plastic when exposed to chemicals.
So, in the end, it's just preference - unless you're cellaring.