For coassociative dg-coalgebras over any field $k$ the answer is positive, because:

 1. Let $C$ be a $\mathbb Z$-graded coalgebra and $D\subset C$ a finite-dimensional ungraded subcoalgebra (of the underlying ungraded coalgebra) of $C$.  Let $D^{gr}\subset C$ denote the graded vector subspace spanned by all the grading components of the elements of $D$.  Then $D\subset D^{gr}$ and $D^{gr}$ is a finite-dimensional graded subcoalgebra of $C$.

 2. Let $(C,d)$ be a dg-coalgebra and $D\subset C$ be a finite-dimensional graded subcoalgebra of $C$.  Set $D^{dg}=D+d(D)\subset C$.  Then $D\subset D^{dg}$ and $D^{dg}$ is a finite-dimensional dg-subcoalgebra of $C$.

Using the observations 1. and 2. and the fact that any ungraded coassociative coalgebra is the union of its finite-dimensional subcoalgebras, one deduces the assertion that any $\mathbb Z$-graded dg-coalgebra is the union of its finite-dimensional dg-subcoalgebras.

Possible generalizations: One can replace a field $k$ by a Noetherian commutative ring $k$ and speak about subcoalgebras that are finitely generated as $k$-modules (instead of "finite-dimensional").  All the assertions remain true.

EDIT: I've realized that the preceding paragraph is problematic for the following reason: given a $k$-submodule $D$ is a $k$-module $C$, the tensor product $D\otimes_k D$ is not a submodule of the tensor product $C\otimes_k C$, generally speaking.  So the very notion of a $k$-subcoalgebra for a commutative ring $k$ is problematic, or at least requires extra care with nonexact tensor products.  So, I retract the preceding paragraph.

One cannot drop the coassociativity condition.  Indeed, even for ungraded coalgebras over a field of characteristic $0$, there is an example of infinite-dimensional Lie coalgebra $L$ having no nonzero finite-dimensional subcoalgebras.  The Lie coalgebra $L$ is simplest described in terms of its dual topological Lie algebra structure (on a pro-finite-dimensional topological vector space): $L^*=\mathfrak g=k[[z]]\,d/dz$, the Lie algebra of vector fields on the formal disk.