In survey research, it is advised that the response rate should be high to avoid self-selection bias. What methods can be used to assess if the data is affected by biases resulting from low response rate,
At a minimum, qualitatively, you can look at what groups were under- and over-represented, look at the areas of consensus/divergence among each of the groups, and in your discussion acknowledge key reasons for bias and the likely effect on the data. You can also model back what the data would look like if you had the subgroup response rates you aimed at, to assess if what you have was really a quant study at all, in the end. You might need to reshape it into something else.
Of course, you will also want to run multivariate analyses to ensure that you are not over- or understating the bias, and it is also wise to state why the low response rate happened and what it in itself says about the issue at hand. You might also conclude that future communication or outreach to the group(s) not as well included should take a different direction, and/or you might conclude that the issue is less salient to them and explain why that could be, as part of your discussion.
If you're lucky enough to have information about characteristics of the target population, and to have collected some of that information about your sample, you could:
1. Run comparisons to see whether your sample deviated notably from the population characteristics. For example, if the target population was 60% female, but your sample was 80% female, then you have evidence that your sample deviates from the population in one potentially important aspect.
2. You could apply weights for these variables to your sample data set, to represent how the results might have looked, had your sample more closely matched with the characteristics of the population.