When is a continuous path stochastic process be representable as diffusion or Ito process? When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
 A: These types of questions are treated in great generality in the book
Rogers-Williams: Diffusions, Markov processes and martingales, Volume 1
One of the great results is due to Dynkin. Let $(X_t)_{t \ge 0}$ be a continuous Markov process whose semigroup is Feller-Dynkin (that is restricts to a strongly continuous semigroup on continuous functions vanishing at $\infty$). Then if the generator of $X$ contains the space of smooth and compactly supported functions, this generator is necessarily a second-order semi-elliptic differential operator and so $X$ is a diffusion process.
A: A simple counterexample is a process that, starting at zero  moves with constant velocity 1 (with probability $1/2$) or with constant velocity -1 (with probability $1/2$).
There are various subtleties here, and the understanding of what a diffusion process is has varied in time and from one author to another. 
In my favorite introductory book on stochastic processes by Wentzell (A Course In The Theory Of Stochastic Processes, originally in Russian), with his choice of definitions, a sufficient condition for a Markov process to be a diffusion is essentially a combination of stochastic continuity and nice behavior of (truncated) mean and variance of transition probabilities.
Update A better example is a strictly increasing nonrandom trajectory that is not absolutely continuous with respect to time.
A: A continuous strong Markov process is a semimartingale if and only if it can be represented as a time-changed Ito diffusion. More generally, a Hunt process is a semimartingale if and only if it is a time-changed jump diffusion. This is a result of Cinlar, Jacod, Protter, and Sharpe, from their truly amazing paper Semimartingales and Markov processes.
