The following two example papers are from biostatistics (and for clinical trials specifically), but might be a useful starting point. I suspect many of the principles will apply, even if the exact analytical approach might differ for different study designs or research paradigms.
They do spell out some criteria for considering whether a sensitivity analysis is useful, or what additional information might apply, such as:
"Is it possible for the proposed sensitivity analysis to arrive at a different conclusion to the primary analysis?" (Morris et al.)
"Will the results change if we take missing data into account? Will the method of handling missing data lead to different conclusions?" (Thabane et al.)
Morris TP, Kahan BC, White IR. Choosing sensitivity analyses for randomised trials: principles. BMC Med Res Methodol 2014;14:11. doi: 10.1186/1471-2288-14-11
Thabane L, Mbuagbaw L, Zhang S, et al. A tutorial on sensitivity analyses in clinical trials: the what, why, when and how. BMC Med Res Methodol 2013;13:92. doi: 10.1186/1471-2288-13-92
Here's an additional article, which has both an overview of reasons to undertake bias analysis and more technical detail on how these can be implemented.
Lash TL, Fox MP, MacLehose RF, et al. Good practices for quantitative bias analysis. Int J Epidemiol 2014;43(6):1969-85. doi: 10.1093/ije/dyu149