For completeness, and because I cannot figure out the general case (cf. my comment below Daniel's answer), let me prove the following: if $f:\Delta\to C$ is a functor which, when restricted to $\Delta_{inj}$ takes values in monomorphisms, then $\lim_\Delta f\to f([0])$ is a monomorphism.
This is simply the following observation : $f([0])\simeq \lim_\Delta f(-*[0])$ (as $f(-*[0])$ is a split-augmented simplicial object), and along thsi identification, the projection map corresponds to $\lim_\Delta (f\to f(-*[0]))$. Now each $[n]\to [n]*[0]$ is an injection, and so this map is $\lim_\Delta$ of monomorphisms, and hence is a monomophism itself.
This seems to use something specific about the combinatorics of $\Delta$.
When $K$ is general, and say has a final object $\infty$, then one can argue similarly: one has a composite $\lim_K f\to f(k) \to f(\infty)$, and thus by some version of 2-out-of-3 it suffices to prove that $\lim_K f\to f(\infty)$ is a monomorphism, but this follows from it simply being $\lim_K (f\to f(\infty))$. My attempt for a general $K$ with $|K|\simeq *$ was to try and prove that the inclusion $K\to K^\triangleright$, where we add a final object, preserves the property of every map being a mono, i.e. that for every $k$, $f(k)\to \mathrm{colim}_K f$ would be a monomorphism - this step is where one would use $|K|\simeq *$, but I didn't manage to prove it.
There is a somewhat easy-to-prove version which is when $K$ is weakly contractible via a zig-zag of functors : if there is a zigzag $id_K \leftarrow h_1 \to h_2 \leftarrow \dots \to h_n$ where $h_n$ is a constant functor, then one can do a similar argument to the one for $\Delta$ and prove the result. I don't remember exactly how often we can expect this: is it the case that any weakly contractible $K$ is a filtered colimit of $K$'s admitting such a zigzag ?