You could rephrase it as: tears are the words YOUR heart can't express, so yes, it is specific. 'Your' refers to the individual reader - each individual reader, even if there are 100's. You could also say: every person/each person, so that it is always specific.

Karen's answer is essentially correct, in that the phrase does mean what she says, but I think I would explain it in almost opposite terms. In this case, "the" tells you that the speaker is talking about hearts in general, NOT a specific heart. You can use the same structure for talking about other types of things or animals in a general way. For example: "The polar bear is a large, carnivorous mammal." (This means the same thing as "Polar bears are large, carnivorous mammals." It is talking about polar bears in general.) "The brain is located inside the skull." (Again, this sentence refers to brains in general. It sounds much more professional than saying "Brains are located inside of skulls.") "At such a great distance, the eyes cannot make out small details." (Eyes in general are not able to see details at such a great distance.) Your sentence could be rephrased as "Tears are words that hearts can't express," but it sounds very clumsy that way. Using the phrase "the heart" to refer to hearts in general is more poetic/literary.