To complement Landon's answer, let me elaborate a bit further.
Causal inference always requires untestable assumptions, the usual ones are absence of direct effects among variables (exclusion restrictions) or absence of unobserved common causes among variables (independence restrictions). For now, let us focus on the violation of these two, but, of course, there are other built-in assumptions as well, such as no selection bias, correctly measured variables, no interference between units and so on.
So the first thing I would point out is that DAGs, as models, have no special status under the "all models are wrong" motto---if you don't write down the implied DAG of your model, your model will still be "wrong" (or, better, "not useful"). Wherr DAGs can really help is to make it easier for you (and your peers) to see where your model could be wrong, and to better pin down where the sources of disagreements are. Then you can assess whether your conclusions are sensitive to that disagreement.
To perform this task, we need tools to derive sensitivity curves of the target quantity of interest in our causal models. Regarding linear structural models, we have just started developing algorithms for making this type of sensitivity analysis automatic and systematic. For instance, take the following example:
Suppose you posit model $G_O$ and obtain that the causal effect of $X$ on $Y$ is identified (and given by the regression coefficient adjusting for $Z$). Now someone challenges you and say that your assumption of no unobserved confounders between $Z$ and $X$ is unreasonable, which leads to the alternative model model $G_A$. In $G_A$, however, the causal effect is not identifiable anymore. So what can you do? Here is where sensitivity analysis comes in.
Instead of point identifying the causal effect, you are going to express the causal effect as a function of other unidentifiable parameters of the model---such as the strength of the unobserved confounders. Then you can see how sensitive your conclusions are to different strengths of that parameter, and resort to outside knowledge and scientific plausibility judgments on those parameters to bound the causal effect of interest.
So the first task we need to solve is to decide whether information on some parameters, say, the strength of confounding between $X$ and $Z$, is sufficient for the identification of the quantity of interest and to find the correct estimand. Once you do that, you can use it to see how sensitivity your estimate is to violations of the zero confounding assumption.
So, back to the example, in $G_A$, can you use that information to bound the causal effect of $X$ on $Y$? The answer here is yes, and we can algorithmically derive the sensitivity curve (if the model were $G_B$, the answer would be no). But suppose you don't have direct external information about the confounders themselves, but you do have some prior studies that give plausible bounds on the causal effect of $Z$ on $X$. Can we use that information for the sensitivity analysis of $X$ and $Y$ instead? Here the answer is also yes. In this way we are building tools to have a disciplined discussion about violations of assumptions on arbitrary causal models (as represented by DAGs).