The key practical value of gate teleportation lies in the fact that it provides a simple and fairly general protocol for reliable and fault-tolerant execution of a wide range of gates.
Depending on the quantum architecture, some gates may be difficult or impossible to execute reliably or fault-tolerantly. The operations that make up the teleportation protocol are often simpler to implement fault-tolerantly than the gate being teleported (see application 1 below). Also, gate teleportation enables us to achieve scalability by performing unreliable operations offline where failures do not restart the entire computation (see application 2 below).
Application 1: Fault-tolerant implementation of non-transversal gates
By the Eastin-Knill theorem, no quantum error correcting code admits transversal implementation of a universal set of gates. While transversality is a simple way to ensure fault-tolerance, it is not the only way. However, most other techniques are ad hoc and do not generalize to many types of gates. Gate teleportation provides fault-tolerant implementation of a large class of gates called the Clifford hierarchy.
The Clifford hierarchy is defined recursively. The first level $\mathcal{C}_1$ is the Pauli group and for $k>1$
$$
\mathcal{C}_k = \{U\,|\,U\mathcal{C}_1U^\dagger \subset \mathcal{C}_{k-1}\}.
$$
All levels of the hierarchy contain potentially useful gates. For example, $\mathcal{C}_2$ is the Clifford group, $\mathcal{C}_3$ contains the $T$ gate and the Toffoli gate and each $\mathcal{C}_k$ contains the controlled rotations used in Shor's algorithm.
The key fact about gate teleportation and the Clifford hierarchy is that to achieve teleportation of $U\in\mathcal{C}_k$ we only need gates in $\mathcal{C}_{k-1}$ (together with measurements and classical control). Consequently, recursive application of the gate teleportation scheme allow us to implement any gate in the Clifford hierarchy. The recursion terminates at $\mathcal{C}_2$ in codes such as the 7-qubit Steane code that implement the Clifford group transversally. In other stabilizer codes it terminates at $\mathcal{C}_1$ or at $\mathcal{C}_2$ if other techniques provide fault-tolerant implementation of the Clifford group.
See this paper for details.
Application 2: Reliable implementation of two-qubit gates using linear optics
KLM protocol enables universal quantum computation using only linear optical elements by using post-selection to implement a non-deterministic two-qubit gate. However, if a gate which succeeds with probability $p$ is used directly in the execution of a quantum algorithm $k$ times then the chance of the overall success of the computation is $p^k$.
We can overcome this exponentially small chance of success by replacing gate application with gate teleportation. The advantage of this approach lies in the fact that failures of the non-deterministic gate do not force a computation to restart. Instead, they merely slow down the offline process of collecting the special states for teleportation.
