It is well known that reflective subcategories of complete categories are complete, and that limits in the subcategory are computed by taking the limit in the ambient category and applying the reflector (however it will act trivially, see the comments). Has this been proven yet for $(\infty,1)$categories? I know that if the ambient $(\infty,1)$category is (locally) presentable, and the subcategory is accessible that this is in HTT, however this is a very special case, and the latter condition is often hard to verify even when dealing with the presentable case. Has anything been worked out on this?

Here's a proof which is certainly overkill, but it has the merit of using references so you can read the proofs in detail. We have $i: \mathcal{C} \subset \mathcal{D}$ a fully faithful subcategory with $r$ a reflector. Step 1. The inclusion $i$ is monadic. Proof: It is clearly conservative, and it preserves and reflects $i$split simplicial objects since $i$ is fully faithful so we can realize the splitting already in $\mathcal{C}$. By BarrBeck (HA.6.2.2.5) the functor $i$ is monadic. Step 2. Monadic functors 'create' limits. Proof: This is the statement of HA.4.2.3.3. where the '$\mathcal{C}$' in that corollary corresponds to $\text{End}(\mathcal{C})$ here, the $\mathcal{M}$ corresponds to our $\mathcal{C}$, and the algebra $A$ corresponds to the monad $i \circ r$. 

