Some interpretations of quantum mechanics — like the Copenhagen interpretation in particular — require the existence of an observer. The role of the observer is a bit mysterious. After all, observers are not needed in classical mechanics. Which physically acceptable interpretations of quantum mechanics do not require the existence of any observer at all? By that, I mean they would continue to make sense even in the absence of any observer. The many worlds interpretations appears to require an observer to subjectively determine which branch is selected.
Adding to the answer by paper_chico,
There are various views including quantum bayesianism, penrose interpretation(objective collapse theory), information-based quantum mechanics, many-world interpretation or relational quantum mechanics according to my knowledge. Personally I like Penrose interpretation, which is based on objective collapse theory and removes the supremacy of observer.
Penrose interpretation: This interpretation assumes that a particle can stay at multiple places at the same time, but since every particle bends space-time around it, it requires some energy to sustain its 'gravitational field' ( space-time curvature,precisely). So, depending upon the energy levels the object can stay at various places.While understanding this, one must rip off the concept of 'wavefunction collapse' from his head. This also explains why macroscopic objects can't exist at multiple places, just by saying that the energy difference is so large, that the objects can't stay there for measurable time. Quoting Wiki:
In Einstein's theory, any object that has mass causes a warp in the structure of space and time around it. This warping produces the effect we experience as gravity. Penrose points out that tiny objects, such as dust specks, atoms and electrons, produce space-time warps as well. Ignoring these warps is where most physicists go awry. If a dust speck is in two locations at the same time, each one should create its own distortions in space-time, yielding two superposed gravitational fields. According to Penrose's theory, it takes energy to sustain these dual fields. The stability of a system depends on the amount of energy involved: the higher the energy required to sustain a system, the less stable it is. Over time, an unstable system tends to settle back to its simplest, lowest-energy state: in this case, one object in one location producing one gravitational field. If Penrose is right, gravity yanks objects back into a single location, without any need to invoke observers or parallel universes. Loosely put, it explains the wavefunction collapse in the following way:
Wavefunctions do collapse, not because of some observer observing it(Copenhagen interpretation), but they will collapse just after the curvature in space-time attains a particular threshold level. If the energy of the particle does not allow the it to stay in different states, it'll collapse to one state.
I'm not an expert in this field, however, I've shared what I know about it. If anyone has better views on this topic, or a deeper insight, please discuss.
Surprisingly, in some of his writings Feynman seemed to lean towards an observer-free interpretation of quantum theory in which objects as small as atoms or particles replace observers by acting as event recorders (memories). Two relevant Feynman quotes are below; Chico in his answer noted several other relevant and related Feynman quotes.
Vol. III, Sec. 3-3 Scattering from a crystal, p.3-9, first full paragraph, talking about whether a neutron will interact with a crystal as a wave or as a particle:
Vol. III, Sec. 3-4, Identical particles, audio version only:
In the first quote, Feynman pretty much says that what counts is whether the event got recorded by an atom, regardless of whether an observer looks at it or not. That's remarkable if you think about it, since it does not even acknowledge the view that a conscious observer must always participate.
Instead, Feynman employed the idea that an atom that is capable of being struck by a neutron is all the "observer" needed to make an individual neutron bifurcate between the two possible paths of remaining quantum (where it leaves no specific information trace is left in the crystal, and the neutron continues as a wave) or becoming classical (where it leaves a small but clear mark on a single atom within the crystal, and ceases to behave like a wave.)
In the second quote, Feynman uses this same concept of information signatures to define the difference between quantum (interfering) and classical situations. Only those situations that fail to leave such an information trace remain quantum. (In current terminology, this would be called the decoherence problem.)
While I really like Feynman's approach to such thought experiments, I must also point out that to the best of my knowledge he never made any attempt to expand such ideas into an explicitly quantified approach. Furthermore, since Feynman clearly did feel that there was such a thing as a universal wave function, and that the concept of wave collapse was "magic" (in a QED footnote I think), it's possible that he was simply making a subtle distinction between "simple" and easily reversible forms of wave functions (the neutron diffracting) and "complicated" wave functions that were far less likely to be reversible (the neutron colliding with one atom). John Bell had a marvelous essay on just that point, in which he showed how very careful one must be when assuming that a "clearly classical" event had no wave representation.
For my part, I like the pragmatic simplicity of Feynman's "did it leave a trace?" rule very much, even if it lacks the formal depth and elaboration of other more abstract frameworks. Quantum events by this rule simply become the ones that don't leave information traces... period. In fact, if you are familiar with Feynman's QED framework and think about it a bit, you will see that Feynman's marvelously accurate and predictive integrals of all possible histories are just the flip side of the same idea.
That is, those many possible histories become significant precisely because no classical event has occurred yet that would contradict them. The absence of classical information traces allows such whispers of worlds that could be to take on a certain degree of measurable reality, if only as vanishingly tiny parts of a wave that is the sum of all such not-yet-forbidden worlds.
 Feynman was very much aware of the conscious-observer viewpoint, since his PhD adviser John Archibald Wheeler was one of its main advocates.
 John Bell, Speakable and Unspeakable in Quantum Mechanics.
For the Penrose interpretation, quoting Wikipedia:
Also, the Wikipedia article on the double-slit experiment slightly mentions about the novel experiments that goes on the corners of the measument problem and the wave-particle duality. For this you should look for "Unsharp particle-wave duality in a photon split-beam experiment", "Simultaneous wave and particle knowledge in a neutron interferometer", "Afshar experiment" and other related experiments and its critiques.
Also, quoting Heisenberg on "Physics and Philosophy":
Which may give you another view for the term observer.
Also, Feynman says on his Feynman Lectures on Physics:
and later on:
When he says, "Nature does not know what you are looking at, and she behaves the way she is going to behave whether you bother to take down the data or not", I think he's ruling out the common sense for observers.
Quantum Mechanics is almost certainly as flawed as every other physical theory that has come and gone. That statement gets on the nerves of a lot of physicists - many of whom think that 'This Time its Different'.
Take Newton's gravity for example. Newton knew that the whole instantaneous transmission of gravitational force did not make sense, yet the theory was wildly accurate and is still used today for 99% of things like space flight. Yet the theory is fundamentally flawed from a conceptual viewpoint.
With QM - where is the problem? It is likely in the transition from the microscopic realm to the macroscopic realm, where the 'large hand wave' called waveform collapse comes into the theory. So getting to the question: Here are some possibilities:
Re: This time its different: http://press.princeton.edu/titles/8973.html
There is always an observer in the sense that someone has to decide what experimental apparatus to construct, what data to collect, what statistics or other multivariate functions of the data to construct from that data, and what theoretical model to compare the data to. That was and is equally true of classical Physics as it is of quantum theory.
There are as many Copenhagen interpretations as there are people who've written about what it is. A recent paper by James R. Henderson, "Classes of Copenhagen interpretations: Mechanisms of collapse as typologically determinative", Studies in History and Philosophy of Modern Physics 41(2010)1–8, doi:10.1016/j.shpsb.2009.08.001 (sorry, I couldn't find a non-paywall version to link to), discusses three people's approaches,
On the second page, Henderson suggests that, for a measurement to happen,
One can take issue with any such classification of the evolving history of the Copenhagen interpretation over the last 80 years, but in relation to your Question I suggest that there is enough to justify a claim that some varieties of the Copenhagen interpretation do not require an observer.
On another front, one of the claimed advantages of the de Broglie-Bohm interpretation is that it is observer-independent in the same sense as classical dynamics (which for some people is desirable enough that they are willing to accept other aspects of that interpretation).
Well if such interpretation existed it would be useless because it would be unable to make predictions for observations by any existing observer.
That said, you correctly noted the many-worlds interpretation. While it needs observers to explain observations, it still has a concept of so-called "universal wave function", a wave function of the entire Universe, that does not correspond to any observer and which undergoes unitary evolution from the very beginning of Universe. The problem with it is that nobody ever can measure it :-) It is just postulated that Universal Wave Function exists, but it is unobservable and at best corresponds to an infinitely distant (in both space and time) "observer".
Moreover. In this work it was shown that universally valid theory is impossible. That is however good a theory is invented the observer could not use it to predict the future of a system that contains himself. This result means that however well a theory predicts future of any observed external objects, the observer himself is excluded from normal application of the theory's laws. As the whole universe inevitably includes the observer himself, this means that no theory can be used predict the future of the universe as a whole because inevitably there is a point in the universe where the theory's laws are broken.