The answer is no. Though nicotine can be absorbed through the skin, not even the highest nicotine contents found in commercial e-juice can cause unwanted side effects. This is because vape juice is not able to be absorbed through the skin quickly enough to have any noticeable effects. The lethal dose of nicotine is estimated to be in the range of 500mg to 1,000mg+ with other side effects and risk of nicotine poisoning occurring at lower doses. Unless you are dealing with concentrated nicotine, getting vape juice on your skin is really nothing to worry about. If you do get e-juice on your skin, don’t panic, just wash it off with soap and water.
It is common for vapers to test flavors by dropping e-juice on their tongue or finger to taste, as well as devices occasionally spitting juice back into the mouths of the users. Getting vape juice in your mouth from these occurrences are almost always in no way harmful. Unpleasant maybe, but definitely not dangerous. For vape juice to be harmful from being ingested orally, it would require large amounts of high nicotine content juice to be consumed; i.e. drinking it. Though you can experience symptoms of nicotine poisoning with smaller doses, in order to cause severe or fatal side effects, someone would have to drink something like a 30ml bottle of 24mg nicotine vape juice which would be a dose of over 700mg of nicotine.
The bottom line; is e-juice dangerous if it makes contact with your skin, or is swallowed? In almost all circumstances, no. Not at all. In order to cause any severe side effects, someone would have to consume a significantly large amount of nicotine. This is a very rare occurrence unless you are working with concentrated nicotine, or if you prefer to drink your e-juice!