Q:

Is there an example that $\mathcal{O}(\mathrm{Proj}\,B)\neq B_0$ for some graded ring $B$?

Is there  a graded ring $B=\bigoplus_{d\in \Bbb N}B_d$ such that  $\mathcal{O}(\mathrm{Proj}\,B)\neq B_0$?

A:

Yes, see for example Vakil's Foundations of Algebraic Geometry, Exercise 7.3.J, which says in part "If $R$ is an $A$-algebra, define a graded ring $S_{\bullet}$ by $S_{0} = A$, and $S_{n} = R$ for $n > 0$. Describe an isomorphism $\operatorname{Proj} S_{\bullet} \simeq \operatorname{Spec} R$." So in this case we would have $\mathcal{O}(\operatorname{Proj}B) = R$ while $B_{0} = A$.
See also Görtz and Wedhorn's "Algebraic Geometry I", Remark 13.7 and the paragraph preceding it: "Let $A$ be a graded ring. The first kind of functoriality is the observation that that we may 'thin out' $A$ and 'change $A_{0}$' without changing the scheme $\operatorname{Proj}A$."
For a more "naturally occurring" example see Global sections of $\mathrm{Proj\,} A[T_0,\ldots,T_n]/\langle T_i T_j\rangle$.

