LWE: does using only a small subspace of the plaintext space influence the security of the encryption scheme? Regarding LWE schemes where the encryption is performed this way:

for $m \in \mathbb{Z}_t$, compute $c = LWE_{\mathbf{s}}^{t/q}(m) = \{ \mathbf{a}, \mathbf{a \cdot s} + m\cdot q/t + e\} \in \mathbb{Z}_{q}^{n}$

(where $\mathbf{s}$ is the secret vector of length $n$, $\mathbf{a}$ is a random vector and $e$ is some noise generated by a discrete Gaussian sampler)

What happens if for a given plaintext modulus $t$, we use only a small subspace of $Z_t$ for our messages? Does it have any influence on the overall security of the scheme?
No, as long as your plaintext space is a subset of $Z_t$, then there is not any negative impact on security.

However, if you use a smaller message space $Z_x\subset Z_t$, you should consider setting the plaintext space to $Z_x$ directly as this could improve the efficiency.