Yes, you could apply the tea tree oil around your eyes which will be absorbed by your eyes surrounding skin to make your eyes skin get smooth and soft because of the inner moisture role. You could have a try. At the same time, you need to find the clean cloth and hot water to do the warm compress to make your eyes feel comfortable and moisture.
Well, at first, you can not confuse tea tree oil with tea oil. The former is extracted form the leaves of the Melaleuca alternifolia, while the latter can be used as a cooking oil. Tea tree oil may be poisonous if it is suddenly swallowed. Tea tree oil can been seen as a new anti-microbial application due to its antiviral and antibacterial properties so that we can wipe them in the skin. As our eyes skin are too sensitive to many irritated substances. So, you are not advised to do that, or some diluted solutions of tea tree oil may be accepted. Besides, they usually can be a remedy to treat bacterial and fungal infections. In addition, tea tree oil's function of skin care seem as a new research field. On safe side, you can first wipe a few on the surface of your hand skin to see if you are allergic to the tea tree oil.
Well, yes, tee tree oil will be fine and in most cases, it will just do a good job for your eyes. And according to some researches, tea tree oil contains a cosmetic product and purcelin oil that can help improve the appearance of scars, stretch marks. And so there are many people who applying tea tree oil to treat black eyes, which is related to injury occurring on eyes. Also, puffy eyes and dark circles under eyes can be treated with tea tree oil too. On the other hand, you should just keep using tea tree oil, and only in that way will it be effective on your eyes.