Does the word "teller" in the meaning of "someone who tells something" mean that the person is quite likely to be a liar?
Teller, in the sense we are talking about, simply means 'one who tells'. There is no suggestion of falsehood. The expression 'truth-teller' is common, and 'teller of truths' almost as common.
In the phrase "teller of tales" it is the tales part that has the suggestion of falsehood, not the teller part.
Teller means someone who counts. Compare it with Norwegian. As an example in that language take the title of a book which was published a few years ago Teller matte? which translates to Does maths count?