As Charles says, you’ll probably be okay if you’re adding liquid fertiliser, but your plants will likely be happier if you stick to tap water instead.
Provided that you are feeding the soil/plants the macro and micro nutrients they need there shouldn’t be an issue. Epsom salt (1tbsp/gallon) every now and then helps with some of the micro nutrients.
Currently I have an aquaponics setup with live fish, worms, snails and a beneficial bacterial colony. For this I cannot use chlorine or chloramine as these chemicals would kill all bacteria immediately. Without bacteria to process fish poop, ammonia would quickly build up to the point of becoming lethal to fish. Chloramine itself is also lethal to fish. Chlorine + fish poop = Chloramine. If you want live animals (including bacteria) in your water, you do have to remove the chlorine somehow, but a tiny amount of vitamin C will neutralise both chlorine and chloramine (http://www.beeradvocate.com/community/threads/vitamin-c-for-chlorine-chloramine-removal.76953/). However, building beneficial bacteria colonies requires specific environmental conditions (oxygen, light, temperature, time, food, substrate). If you can’t reliably provide these conditions 24/7, you’re more likely to be left with only bad bacteria/mould as it tends to outcompete the good bacteria outside these specific conditions. This bacteria/mould can make your plants sick in the long term, especially so if they are nutrient deprived (from persistent use of distilled water).
I also trialled adding a product based on chloramine (marketed as “Pythoff”) to the hydroponic setup, which is a chemical sometimes used in tap water (chlorine + ammonia), to stimulate root growth and it worked quite well. It basically works by giving the plant roots a sterile environment to grow in, removing both good and bad bacteria. Removing chlorine from tap water is absolutely not necessary for plant growth and will simply encourage bacteria and mould growth, for better or worse. Depending on the environment, you may not want to encourage this type of growth in your garden. If your apartment is very humid, it might be better to leave the chlorine in the water to help fight unhealthy microbeasties.
I recently grew plants hydroponically in distilled water. It worked fine, but I had to manually measure and re-add the required minerals for the plants to be healthy. If you can’t or don’t want to carefully re-add minerals I recommend you avoid distilled (or reverse osmosis or similar) water for plant survival.
It may also be worth checking that your water is actually being treated with chlorine and not chloramine (check with your local water supplier). In the case of the latter, simply leaving it out may not be enough for the chemicals to break down/off-gas (this is why chloramine is used, as it tends to stick around).
Minerals for your plants can be supplemented via plant food, rock dust, azomite or greensand if needed or through the soil/potting mix of the plants unless you are using a sterile potting medium. Provided that you repot every now and then you should be fine.
Like everything else it depends on the plants and their needs. Some believe you should provide as many trace minerals as possible and let the plant decide what it needs.
Just over a hundred years ago, in 1915, a chap called M.C. Merrill did some experiments on this (that link may be paywalled, sorry) and found that, while plants would grow with distilled water, it was not a great choice. (If you prefer to read the very lengthy report of that study in its entirety, it’s available for free in the Annals of the Missouri Botanical Garden.)
Can I use “distilled” water to water plants? My apartment is very humid, and I regularly use a dehumidifier. Is it OK to use the resulting water to water my indoor plants? Since it’s basically