Yes, a many-qubit state (even if it entangled) can be teleported by teleporting each qubit separately using one (perfect) Bell pair and two bits of classical communication. (This is the idea of quantum teleportation: The qubit, including all of its quantum correlations, is teleported.)
This can be seen by observing that teleportation of a qubit implements the identity channel on the qubit, i.e., it maps any state $|\alpha\rangle$ to itself (while preserving the phase!). More precisely, teleportation (like any physical map) is a completely positive trace preserving (CPTP) map which is of the form $\mathcal E(\rho)=\rho$, i.e., it also preserves coherences. Several independent teleportations thus implement the identity on $N$ qubits, i.e., they preserve any states (including entangled ones).
On the other hand, if there is noise (i.e., the entangled states are not perfect) you will need to use an encoding/decoding scheme. For general noise (however small it is), this will only work asymptotically, i.e., you will need to teleport a large number $N$ of qubits using $cN$ imperfect Bell pairs (with $c>1$ some constant), and in the limit $N\rightarrow\infty$, the teleportation will work perfectly given $c$ is not larger than the quantum capacity of the channel.