There are a bunch of good answers here that are clearly showing that information beyond genetics can be encoded in DNA and those are indeed answers to the question as it is phrased. But the question made me wonder about DNA, not as written language, which is implied by the asker, but rather as "spoken" language - I doubt that any of us would find the coding discussed so far a suitable and satisfactory substitute for what we call language in our daily lives. So I think the we need to answer (at least I want to answer), "can DNA be used by a species just the same way we use spoken language or sign language". Dynamic, interactive, adaptive symbolic code capable of communicating, on a contemporaneous basis, information, emotion, etc.
Can DNA have a syntax? A grammar? Yes, we know genetic coding already has both syntax and grammar so those tools exist for our DNA language.
We need a organ to "speak" our language and an organ to "hear" our language. We can certainly imagine such organs. Our cellular machinery can already perform the tasks of writing (copying) and reading DNA strands. But our imagined language organs will need to be much more nimble. Our speech organ will need to assemble DNA strands based on inputs from our nervous system; Building DNA strands representing words and sentences and then emitting them to the medium between the speaker and the listener (air, water, ??). Such an organ would probably need to emit many copies of our sentence since our words are traveling in matter packets (not energy waves) and cannot be "heard" until the molecule of our sentence arrives at the receivers hearing organ. Sending just a single sentence molecule would result in a very poor chance of ever being heard! We must puff our sentence out a thousand times simultaneously like pollen from a flower on the wind. Of coarse, our words and sentences will need to contain a grammar similar to a time stamp. If I say "Hi, How are You?" 1000 times, and the recipient receives 120 of those "Hi, How are You's?", I don't want him to "hear" all of them. I need to give him a grammar to hear the first "Hi!" and filter out all the rest as noise.
So, on the "hearing" end of the conversation, we need an organ filtering the media between us and the "speaker" scanning for floating bits of DNA. Assuming we are on a planet that uses DNA for genetics, not just language, there is going to be a lot of DNA genetic material floating about. So our DNA language grammar must make a clear distinction between a word or sentence and genetic DNA which is just noise to be filtered out and ignored. If this filtering happens at the biological level of the organ itself, then calling it a "grammar" may be misleading. For instance, the organ might only bond with DNA strands that begin with a complex 100 nucleotide sequence. Any DNA strand that does not have this telomere at the beginning/end does not bind and is not "heard". Either way, biological or grammatical, the hearing organ responds to the coding in the captured and read DNA strand by activating the hearer's nervous system. Just as our ears do for us.
So now that we have imagined the machinery for a species to have DNA molecule based language, we can imagine some consequences and advantages...let's go scifi.
In the case where we use the biological filters on our organs (complex telomere tag is required or molecule is not read because it wont bind) any individual who suffered a mutation that either resulted in there speaking organ coding the wrong telomere or there hearing organ failing to bond to the correct telomere would be rendered either "mute" or "deaf".
Conversations would be subject to very different rules when your words can be blown with the wind and be heard miles away! Or when no one up wind of you can hear you no matter how loud you shout. Also, since your words and sentences are subject to the vagaries of the wind or just random distribution by Brownian motion, our language and our nervous system will need to evolve to deal with the fact that our sentences will often not arrive with the same timing, or even in the same order, that we spoke them in!
And for that matter, what is a shout? Producing 100,000 DNA sentence copies at a time? Making a whisper the creation of just a few copies?
As this species evolves, will they evolve some level of encoding to enable private conversation? Perhaps the hearing organ can hear a DNA molecule starting with any telomere sequence, but it can hear (bind) only one possible telomere at a time. So most conversations happen with telomere Alpha. But when two or more individuals want to talk voce sotto, they switch over to telomere Lambda or Omicron. There words are still arriving at the hearing organs of others, but are ignored by all who are not "listening" for Omicron.
What is a recording device on this planet? Can biological microbes be engineered to reproduce / record conversations by replicating the DNA that makes up the sentences and words?
Since words are matter, not energy, they potentially have a lifetime far beyond what we are used to. You could capture the last words of your mother in a jar, and hear them again...and again....and again for years - until the last of the molecules of that sentence were used or lost. NOT just a recording - the actual voice of your mother, her spoken words from her "mouth", unchanged just as if she was there with you in the room.
Specters of conversations might linger in buildings for years or generations. Imagine closing a door in Philadelphia's Liberty Hall and dislodging from a dusty rafter a few sentences between Ben Franklin and Thomas Jefferson arguing over a paragraph in the Declaration of Independence! As those sentences/molecules drifted into your "ear", you would hear Franklin and Jefferson - their words in their voices - unchanged from the moment they spoke them 250 year ago. The occupations of archaeologist and historian would be something quite different.
Finally, since DNA is still the mechanism for genetic information encoding and transmission, can these creatures "speak" genetic information? For instance, could a lover "whisper" a hormone that caused the object of there affection to feel flush? Or an RNA strand, inhaled by the subject which caused the production of the hormone by his/her own limbic system. Or could a rival encode and surreptitiously breath a virus onto his victim? Or, the most insidious end to this line of reasoning...could a sentence itself be encoded in such a way that hidden within it were the very instructions that caused the hearer's cells to produce the virus in their own body?! Just like sticks and stones, on such a world, words can kill you!