I am a Duolingo user, currently working on the Spanish course. I do find it to be beneficial, but it does have its limitations. A few that I've noticed are:
As has been mentioned, the sentences used are often not very practical or even sensical; how often do I need to tell someone "I am a penguin" or "that is surely my elephant"? However, some may find that this makes the lessons interesting and memorable.
The listening exercises for many courses use a computer voice, so while you get some practice identifying words in speech, sometimes the computer pronunciations are unclear, and you also don't get the native speaker intonation.
The explanations of some vocabulary and grammar are confusingly incomplete, for example, in the Spanish course, "gustar" is translated as "to like", with conjugations "yo gusto", "nosotros gustamos", "ellos gustan", etc. So you would definitely have to go to some other source to get a better understanding.
There is an inherent limitation in relying on translation to/from another language, as Duolingo does exclusively. One speaks most fluently when one no longer is mentally translating from the native language or another language. Words in one language often do not have an exact equivalent in another, and Duolingo does take this into account to some extent by translating certain phrases as a unit. But, while translations are good for beginners, in order to be fluent you have to think directly in the target language, which is not well encouraged by Duolingo; learning through context and real-world association are better for that.
There is also limitation is using isolated words, phrases, and sentences. While knowing how to say "I do not reject food" is useful in some circumstances, it's still just an isolated sentence without context. I personally find it easier to learn words and grammatical constructions in context, such as a story, a conversation, a letter to a friend, etc. This gives more perspective on usage, pragmatics, idioms, etc.
All that being said, there are a lot of good things about Duolingo; but the question only asked for the drawbacks.