There are several instances where essential oils are not a good option to be interested in. At least it requires some thoughts and considerations.
Pets
If you have indoor pets roaming around your house, you can forget about any sort of aroma-spray adding ambience to your rooms or inhalations. The animals are usually repelled by the sensations - dogs and cats have a much more powerful sense of smell than humans, after all. They may start to act strangely, becoming restless, fearful, avoiding contact. We don't want that.
If you want to use essential oils while having a little adorable beastie, you can only use them directly on your body in a form of a massage. Remember not to stroke your pets at least after an hour of applications. We use hands to apply the oils. You can have one dedicated room where your pets will not go, eventually.
Knowledge
If you want to know aromatherapy superficially, because it is trendy, you may want to stick to just lavender and patchouli oil as they are safe to use on many occasions. Otherwise, you need to dive deep into the subject to not harm yourself and others. Essential oils are very potent substances and some of them require special attention, knowledge and experience.
Of course there is a question of what sources there are available. If you want to know anything about particular essential oils on websites not dedicated to aromatherapy you will find a lot of misleading information. The articles are written by 'copy paste' and rearranged slightly. According to those texts all essential oils are good for all purposes. This is far from the truth.
Danger
There are potential dangers in using essential oils, several of them. We do not know the origin and exact ingredients of an oil. It can be adulterated, even if the intentions of the seller are pure and sincere. We simply would never know and would never be sure.
The plants have different qualities year to year, and the power of an essential oil may be quite different even if extracted from the same plant and field. The field may be near a farming field where pesticides have been in use. Only after months or years we may notice something odd, and never know why.
Not having enough knowledge about different oils and how to use them, as mentioned above can be quite problematic. You need to dedicate a lot of time to know how to handle them. Essential oils should not be used by inexperienced person.
In fact, after some years of using essential oils, I have returned to more traditional recipes for some applications, where the ingredients are simple and safe. I have noticed that sometimes they work as good as the essential oils would, without a danger of burning your skin or becoming allergic.
Price
Some believe that the more expensive the oil, the more pure it will be. Not quite. As mentioned above - we never know what is in the bottle. There are other cheaper products that get you a lot of required effect. Especially dried herbs.
As an example - mix baking soda, coconut oil and sea salt for a wonderfully soothing bath. I know it will not have the wonderful smell, but relaxing in it is second to none.
There are certain kinds of people who tend to go over the top with things. The prices of essential oils can go crazy for a small bottle. It became a trend in some circles - having the best, for the highest price. If you are going to be a professional aromatherapist, and have some money to spend - that is fine. Otherwise, do not bother, you will have a similar if not better experience with a herbal infusion and other natural or more easily available ingredients.
