Response time of scheduling a DAG where each vertex is a task
Found it! This problem is NP-complete when the goal is to minimize total execution time.

The scheduling problem (P1) is the following. We are given
(1) a set $S = \{J_1 , \ldots , J_n\}$ of jobs,
(2) a partial order $\prec$ on $S$,
(3) a weighting function $W$ from $S$ to the positive integers, giving the number of time units required by each job, and
(4) a number of processors, $k$.

Please see the following paper, which immediately discusses it from the introduction onward:
J.D. Ullman. Np-complete scheduling problems. Journal of Computer andSystem Sciences, 10(3):384 – 393, 1975.
There are a few other cases that may be of interest that are discussed in the paper:

When $\prec$ is empty (no edges in the DAG)
(P2) When $W(J_i) = 1$ for all $i$.
(P3) When $k = 2$ and $W(J_i) \in \{1, 2\}$ for all $i$.

You can show 3-SAT reduces to (P2) through and intermediate problem (P4) that Ullman discusses. Then it is clear that (P2) reduces to (P1).