Does hyperconjugation only arise from C–H bonds?
No.
Hyperconjugation is the donation of electrons from a one orbital to another, where at least one of those orbitals is a bonding orbital, and there is a net overlap between the orbitals.
A typical first example taught of hyperconjugation is that which occurs in the conformations of simple alkanes (e.g. ethane). When ethane is in the staggered conformation, electrons from the $\sigma_\ce{CH}$ bonding orbital are donated to the properly aligned $\sigma_\ce{CH}^*$ antibonding orbital of the adjacent carbon atom. The interaction of these two orbitals, the $\sigma_\ce{CH} \to \sigma_\ce{CH}^*$ lowers the energy of those electrons, and thus lowers the energy of the molecule.
Next up is to look at a longer alkane chain (e.g. butane), which will have $\sigma_\ce{CH} \to \sigma_\ce{CH}^*$ interactions like in ethane, but will also have additional interactions involving the $\sigma_\ce{CC}$ and $\sigma_\ce{CC}^*$ bonding &amp; antibonding orbitals. Depending on which conformation of butane you are considering, it is possible to have $\sigma_\ce{CC} \to \sigma_\ce{CH}^*$, $\sigma_\ce{CC} \to \sigma_\ce{CC}^*$, and $\sigma_\ce{CH} \to \sigma_\ce{CC}^*$ interactions.
Continuing in a more general sense, these interactions are not limited to $\sigma$ bonding orbitals. In carbocations, which in principle have an empty p orbital ($\ce{C}_\mathrm{2p}$) on the electron-deficient carbon, there can be hyperconjugation in the form of $\sigma \to \ce{C}_\mathrm{2p}$. (Note that I didn't specify the atoms involved in the $\sigma$ bond; as long as the atoms form a stable $\sigma$ bond, this hyperconjugation interaction can occur, though how stabilizing it is will depend on the size of the orbitals involved.)
There are a variety of hyperconjugation interactions that can occur:

$\sigma \to \sigma^*$
$\sigma \to \pi^*$
$\sigma \to$ empty hybridized orbital
hybridized orbital $\to \sigma^*$
$\pi \to \sigma^*$

For example, in protonated formamide ($\ce{H-C(=O)-NH3+}$), there is hyperconjugation in the form of $\pi_\ce{CO} \to \sigma_\ce{NH}^*$ and $\sigma_\ce{NH} \to \pi_\ce{CO}^*$, as well as $\ce{O}_\mathrm{2p} \to \sigma_\ce{CN}^*$.