Is it correct, according to quantum mechanics, to say that an observer can get entangled with the measured system? Entanglement applies to any physical system, observers included. However, what one typically means by the observer being entangled with the measured system seems not to comply with the requirements for such a phenomenon to take place. For example, splitting a spin-zero diatomic molecule will produce a pair of spin-entangled atoms as a result of angular momentum conservation. On any axis the spins of the atoms will be found to be anticorrelated. On the other hand, when an atom passes through a Stern–Gerlach device and is detected on a screen we need to apply momentum conservation to the entire experimental device, not only to the photons that ultimately arrive on the observer's retina. Therefore it seems to me that there can be no entanglement between the measured atom and the observer.
 A: If a quantum mechanical particle's state is completely indeterminate - that is, if its probability of being found in state A is 50% and its probability of being found in state B is also 50% - then the observer, his/her measurement apparatus, and everything else that changes depending on the outcome of a measurement of the particle's state, becomes entangled with the particle when a measurement is performed.  
From the Many Worlds perspective, the measurement causes the universal wave function to split into two independent components corresponding to the two possible outcomes of the measurement.
However "entanglement" of the observer and the particle is not something the observer detects.  It would have to be detected by a second observer who can perform subsequent measurements on both the first observer and the measured particle,  and can repeat his measurements multiple times on identically generated particles and (first) observers.
Edit: Though it's effectively impossible to create multiple observers all in the same quantum state, it might be possible to design an indirect experiment that can be done using a single observer to demonstrate that the observer is entangled with measured particles.  I think this would boil down to a variation on Schroedinger's Cat, with the cat being the observer.
A: Let's say that we have a qubit. Lets also say we have a measuring device (or 'observer') which will measure the qubit in the computational basis. Our measuring device obeys quantum mechanics and returns a result $M$.
If we initialise our qubit in the state $\left|0\right\rangle$ and then measure it with our measuring device, the state of the measuring device becomes $\left|M= 0\right\rangle$. Similarly, if we initialise our qubit in the $\left|1\right\rangle$ state and meausre, our measuring device will end up in state $\left|M=1\right\rangle$. So far what we have described is just the definition of a measuring device which obeys quantum mechanics. If our measuring device behaved any differently to what we have just described, then either it does not obey quantum mechanics, or it is not a measuring device (eg. if, when we measured the $\left|0\right\rangle$ state, our device sometimes returns $\left|M=1\right\rangle$, it would not be a proper measurement device as it gives us wrong information).
Lets say that the measuring device is initialised to a state $\left|I\right\rangle$. We can therefore write the effect of the measurement process as follows:
$\left|0\right\rangle\left|I\right\rangle \rightarrow \left|0\right\rangle \left|M=0\right\rangle$
$\left|1\right\rangle\left|I\right\rangle \rightarrow \left|1\right\rangle \left|M=1\right\rangle$
Now, since quantum mechanics is linear, we can also have a qubit in a superposition state $\left|\psi\right\rangle = a\left|0\right\rangle + b\left|1\right\rangle$, where $|a|^2 + |b|^2 = 1$. What happens when we do a measurement on this state? Well, since quantum mechanics is linear our measuring device enters into a superposition:
$(a\left|0\right\rangle + b\left|1\right\rangle)\left|I\right\rangle \rightarrow a\left|0\right\rangle \left|M=0\right\rangle + b\left|1\right\rangle\left|M=1\right\rangle$
Notice now that the joint state of the qubit and measuring device is an entangled state. There is plenty of discussion by physicists about what qualifies as a measurement, but this is what people mean when they say a system becomes entangled with the measurement device. To satisfy your actual question, you can replace 'measurement device' with 'observer' without affecting the argument.
