Worm refers to a serpent and tongue refers to the power of words/speech. It is a metaphor describing how the speaker's words could "bite" a person like a snake. It is not a reference to figurative meanings of the word "snake" such as a deceitful person, e.g. snake in the grass.
The term's origins have nothing to do with Lord of the Rings or Harry Potter. It existed long before either fiction was created.
Gunnlaugr Ormstunga was a famous Icelandic poet known as Gunnlaugr Wormtongue for his scathing and derogatory poetry.
Most televangelists, used-car salesmen, politicians, hookers, drug dealers and lawyers have worm tongues and have to the power to seduce people with their voices.
The origin of the worm tongue is most-likely the character created by J.R.R. Tolkien in “The Lord of The Rings” trilogy.
Another example, the wormtongue persuades the 'friend' that doing an activity (asking a person out on a date, trying out for a new activity, etc.) won't work, will cause problems, is actually a bad choice, etc. when the wormtongue really wants the 'friend' to not succeed. This is done for a variety of reasons but they all involve some sort of hypocritical gain on the part of the wormtongue.
On the web, a lot of chatrooms and blogs have wormtongues that appear to be on one side of an issue or else neutral but in reality are very biased and working for the other side. This is especially true in the case of religious or political issues such as global warming, Bush v. Obama, etc.