Can fire cider go bad with rotten fruit?

I don't know what you might be incubating in your fire cider, but it is possible that some wild yeasts have taken up residence. Any of the ingredients can bring those with them- except the vinegar. However, vinegar mother does live on alcohol, which it converts to acetic acid. Yeasts produce alcohol as a byproduct, so the first step in making vinegar is to make alcohol- and that may be what is going on in your jar. A jar of vinegar in the making does have some odd off smells at times. Alcohol itself doesn't "go bad" in the usual sense, but as it sits- the flavor profile can change. And all yeasts don't produce alcohol you would care to drink, either. Nasty alcohol will also make nasty vinegar. Those secondary flavors produced by yeast are the reason wine and beer producers are pretty picky about their yeast strains. I suggest you take a look at the liquid in your bottle, and see if you can spot any signs of a fermentation going on inside- bubbles would be a major clue. If this is so, then you can let it finish the job and see what you end up with. For the fermentation period, it will need to have an airlock of some sort- or you will want to burp the bottle daily. Once you see no more bubbles, you can give it a taste and decide what you think. Nothing in it will hurt you, though it may insult your taste buds. You may also get a nice "mother" if you opt to let her have air and space to convert the alcohol to vinegar. That will also change the flavor profile as well- but if it tastes good going in, it should taste fine as vinegar. I hope this answer makes some sense. I make both beer, wine and vinegar- and it's an adventure every time. Additional flavorings are what have given us apertifs, tonics, and all the variations of beer and wine, after all. You may have just discovered one unique to you.