Let $R$ be a discrete valuation ring with residue field $k$, an algebraically closed field of characteristic zero and $\pi:X\to \mbox{spec}(R)$ a smooth, projective family of surfaces. Denote by $X\_0$ the special fiber in the family. Suppose there exists a curve $C$ in $X\_0$ which does not lift as a subcurve of $X$, meaning there does not exist a subcurve of $X$, flat over $R$, which restricts to $C$ on the special fiber. 


Is it possible that for the infinitesimal, first order deformation $X\_0'$ of $X\_0$ along spec$(R)$, the curve $C$ lifts to a local complete intersection subcurve of $X\_0'$? If so, is there any general condition when this does not happen?



Families of smooth projective varieties over dvr
That can happen. Here is one source of examples. Consider $\mathbb{P}^3\_k = \text{Proj}\ k[s,t,u,v]$. Let $d\geq 4$ be an integer, and consider hypersurfaces $X\subset \mathbb{P}^3$ of degree $d$, i.e., $X$ is the zero locus of a degree $d$ homogeneous polynomial, $$F(s,t,u,v) = \sum\_{\alpha+\beta+\gamma+\delta=d} c\_{\alpha,\beta,\gamma,\delta} s^\alpha t^\beta u^\gamma v^\delta.$$ The parameter space for these hypersurfaces is the projective space $$P = \mathbb{P} H^0(\mathbb{P}^3,\mathcal{O}(d)) = \{ [c\_{\alpha,\beta,\gamma,\delta}]\_{\alpha+\beta+\gamma+\delta=d} \} .$$ For every line $L\subset \mathbb{P}^3$, e.g., for the line $C=\text{Zero}(u,v)$, the parameter space $P\_L\subset P$ for hypersurfaces that contain $L$ is a codimension $d+1$ linear subspace, e.g., $$ P\_C = \text{Zero}(c\_{\alpha,\beta,0,0})\_{\alpha+\beta=d}.$$ This parameter space contains points parameterizing smooth hypersurfaces, e.g., for $L=C$, one example is $$ X\_0 = \text{Zero}(us^{d-1}+vt^{d-1}+u^d+v^d),$$ (this works when the characteristic is prime to $d$ and $d-1$, but there are other examples that work for the excluded characteristics). 


In particular, since there is only a $4$-dimensional family of lines $L$ in $\mathbb{P}^3$, the union over all $L$ of $P\_L$ is a closed subvariety of $P$ of codimension $d+1-4$. Since $d\geq 4$, this codimension is $\geq 1$. The complement of this proper closed subvariety is a Zariski dense open subset $U$. Thus, there exists a point $[b\_{\alpha,\beta,\gamma,\delta}]$ in $U$ that parameterizes a smooth hypersurface $X\_\infty\subset \mathbb{P}^3$ that contains no lines. (If need be, there are explicit equations for such a hypersurface $X\_\infty$.)


Now choose $[c\_{\alpha,\beta,\gamma,\delta}]\subset P\_C$ parameterizing a smooth hypersurface, e.g., the hypersurface $X\_0$ from above. For the DVR $R=k[z]\_{\langle z \rangle}$, for every integer $r\geq 1$, form the homogeneous polynomial
$$
F\_{r,z}(s,t,u,v) = t^r(us^{d-1}+vt^{d-1}+u^d+v^d) + z^r\sum\_{\alpha+\beta+\gamma+\delta = d} b\_{\alpha,\beta,\gamma,\delta} s^\alpha t^\beta u^\gamma v^\delta.
$$
The zero scheme of $F\_{r,z}$ in $\mathbb{P}^3\_R=\text{Proj}\ R[s,t,u,v]$ is a hypersurface $X$ in $\mathbb{P}^3\_R$ such that $X\_0$ contains $C$, and to order $r-1$ $X$ equals $X\_0 \times\_{\text{Spec}\ k}\text{Spec}\ R/\langle z^r \rangle$, so contains $C\times\_{\text{Spec}\ k}\text{Spec}\ R/\langle z^r \rangle$. However, the generic fiber contains no line. 


Note that $F\_{r,z}$ is the base change of $F\_{1,z}$ by the finite flat morphism of DVRs $\text{Spec}\ R \to \text{Spec}\ R$ obtained from the finite flat ring homomorphism $z\mapsto z^r$. Of course there are other examples that do not factor through such a base change. However, this shows a conceptual difficulty with your general question: you will need to rule out all examples that factor through a base change. 


One way to do this would be to demand that the Kodaira-Spencer map $$\text{Hom}\_k(\mathfrak{m}\_R/\mathfrak{m}\_R^2,k)\to H^1(X\_0,T\_{X\_0/k}),$$
is nonzero. Then the image of the Kodaira-Spencer map is a $1$-dimensional subspace. When $C$ is a local complete intersection curve, the short exact sequence of sheaves of differentials,
$$
0 \to \mathcal{I}\_{C/X\_0}/\mathcal{I}^2\_{C/X\_0} \to \Omega\_{X\_0/k}|\_{C} \to \Omega\_{C/k} \to 0
$$
gives rise to a long exact sequence of groups $\text{Ext}^m\_{\mathcal{O}\_C}(\bullet,\mathcal{O}\_C)$, one part of which is a connecting map
$$
H^1(C,T\_{X\_0/k}|\_C) \to \text{Ext}^1\_{\mathcal{O}\_C}(\mathcal{I}\_{C/X\_0}/\mathcal{I}\_{C/X\_0}^2,\mathcal{O}\_C).
$$
If the image of the Kodaira-Spencer map under this connecting map is nonzero, then there is no flat lift of $C$ to the first-order deformation $X\_0'$.


