stochastic calculus - Distribution of running maximum of a local martingale

No. It is true that $mathbb{P}(X^*_T>beta)=O(beta^{-1})$, but you don't have a`little-o' bound. In fact it fails, and $beta,mathbb{P}(X^*_T>beta)$ converges to a strictly positive value, precisely when $X$ fails to be a martingale.

If $S$ is the first time at which $X$ hits $beta>x$ then continuity gives
$$
X_{Swedge T} = beta 1_{{X^*_T>beta}}+1_{{X^*_Tlebeta}}X_T
$$
Take expectations, and use $mathbb{E}[X_{Swedge T}]=x$, which follows from the fact that the first term is a local martingale stopped at time $S$, so is bounded (and hence a proper martingale).
$$
x=beta,mathbb{P}(X^*_T>beta)+mathbb{E}[1_{{X^*_Tlebeta}}X_T].
$$
The final expectation converges to $mathbb{E}[X_T]$ as $beta$ goes to infinity, by monotone convergence. This gives
$$
lim_{betatoinfty}beta,mathbb{P}(X^*_T>beta)=x-mathbb{E}[X_T].
$$
Now, it is a well known result that if $X$ is a nonnegative local martingale and $X_0$ is integrable then it is a supermartingale, so $mathbb{E}[X_T]lemathbb{E}[X_0]$, and equality holds precisely when it is a martingale over the range $[0,T]$. So, in our case, $mathbb{P}(X^*_T>beta)=o(beta^{-1})$ exactly when $mathbb{E}[X_T]=x$ and $X$ is a martingale over the range $[0,T]$.

An example when solutions to your SDE fails to be a martingale is $sigma(x)=x^2$, $dX=X^2,dW$. The solution to this SDE can be written as $X=1/Vert BVert$ for a 3-dimensional Brownian motion $B$ started from the point $(x^{-1},0,0)$. You can calculate $mathbb{E}[X_t]$ and determine that it is decreasing in $t$, so $X$ is not a martingale - just a local martingale. This example appears in Roger's & Williams book Diffusions, Markov Processes and Martingales as an example of a local martingale which is not a proper martingale.