Physiological Significance of Superposition To what extent is the concept of superposition appropriate to physiology? In other words, can organisms "actually" exist in a state of superposition? 
 A: No they cannot. Any living system is an incredibly complicated, interacting quantum system. The countionous interaction (noise, heat bath, etc.) between particles makes it impossible for a coherent state to exist. However this is only true for the whole organism, on the level of single proteins, or other biomolecules quantum coherence may play an important role. 
(Search for "quantum biology" for more. You might find some esoterics, but this is a perfectly legit and exciting area of physics.) 
A: Superposition stays until not observed. Organism as a whole can stay in such state (like both dead and alive Schrödinger's cat) until not observed, which has a negligible possibility in the physical world (physical-scales). In the quantum world (quantum-scales), it is possible. 
A: Some of the other answers and comments have pointed out decoherence as a major obstacle for achieving superposition at macroscopic levels (which is absolutely correct, specially for the whole organism). However, this might be interesting: according to this recent paper, Nature could have invented a way to maintain coherence for sufficiently long periods of time in biological systems such as the human brain. The key concept here is the nuclear spin, which is a quantum property related with the decoherence time: the lower the nuclear spin, the less the nucleus interacts with its sorroundings, then the less quickly it decoheres. In particular, the author proposes that the nuclear spins of something called Posner molecules (clusters of calcium and phosporous atoms), which have been found in the human body, could remain entangled for long periods of time, serving as neural qubits, allowing the brain to store quantum information. All of this is summarized in the article's abstract:

The possibility that quantum processing with nuclear spins might be operative in the brain is proposed and
  then explored. Phosphorus is identified as the unique biological element with a nuclear spin that can serve as
  a qubit for such putative quantum processing - a neural qubit - while the phosphate ion is the only possible
  qubit-transporter. We identify the “Posner molecule”, Ca9(PO4)6, as the unique molecule that can protect the
  neural qubits on very long times and thereby serve as a (working) quantum-memory. A central requirement for
  quantum-processing is quantum entanglement. It is argued that the enzyme catalyzed chemical reaction which
  breaks a pyrophosphate ion into two phosphate ions can quantum entangle pairs of qubits. Posner molecules,
  formed by binding such phosphate pairs with extracellular calcium ions, will inherit the nuclear spin entanglement. [...] Multiple entangled Posner molecules, triggering non-local quantum correlations of neuron firing rates,
  would provide the key mechanism for neural quantum processing. Implications, both in vitro and in vivo, are
  briefly mentioned.

Here is an excellent popular-level explanation of Fisher's proposal and investigation. I know little to nothing about the plausibility of such hypothesis, but you can judge by yourself.
