Let $X$ be a vector space equipped with a norm $p$ and a seminorm $q$. Denote the completion of $X$ with respect to $p$ with $X_p$ and with respect to $p+q$ by $X_{p+q}$. Then the induced map $\iota : X_{p+q} \to X_p$ is well-defined and continuous but not necessarily injective as can be seen in analogy to this answer to Kernel of the Extension of a Bounded Linear Operator by taking $X = l^2$, \begin{equation} p(x) = \sqrt{\sum_{n = 1}^\infty \left( \frac{x_n - x_{n+1}}{n} \right)^2}, \qquad q(x) = \sqrt{\sum_{n = 1}^\infty \left( \frac{x_n}{n} \right)^2} \, . \end{equation} Then setting $x^m_n = m/(m+n)$ we have $\lim_{m \to \infty} p(x^m) = 0$, $x^m$ being Cauchy in $q$ and $\lim_{n \to \infty} q(x^m) = \sqrt{\pi^2/6}$ such that the equivalence class of $x^m$ in $X_{p+q}$ lies in the kernel of $\iota$ while being nonzero in $X_{p+q}$.

On the other hand we can take the Schwartz space $X = \mathcal{S}(\mathbb{R})$ and let $p = \lVert \cdot \rVert_{L^1}$ and $q = \lVert \cdot \rVert_{L^2}$. Then $\iota$ is certainly injective because any sequence $f_n$ in $\mathcal{S}(\mathbb{R})$ with $\lim_{n \to \infty} \lVert f_n \rVert_{L^1} = 0$ has a subsequence converging to zero almost everywhere such that if $f_n$ is Cauchy in $L^2 (\mathbb{R})$, the corresponding limit $f$ in $L^2 (\mathbb{R})$ (which exists by completeness) has to be the zero function. Thus $\lim_{n \to \infty} \lVert f_n \rVert_{L^2} = 0$ as well proving that the equivalence class of $f_n$ is zero in $X_{p+q}$.

In the latter example we have much more structure: $X$ is a reflexive, complete and metrisable nuclear space and $p$ and $q$ are continuous norms on $X$. But I suspect that these conditions are not sufficient.

What are some necessary and sufficient conditions for $\iota$ to be injective?