Handling missing data for participants who have not completed any standardised measures and have only provided demographic answers When managing missing data, how many questions should participant have completed, at a minimum, before imputing the remainder of their missing data?
For example, a number of my participants only completed demographic variables, but failed to complete any of standardised assessment measures (so no dependent or independent variables were completed). It seems illogical to include these people in multiple imputation (or other method) as they are essentially missing >90% of their data, but would need to justify this and don't know how.
I completed Little's MCAR test and data was actually MCAR and is also missing monotonically. I also completed Chi-square analyses to look for demographic differences between those who completed at least one standard measure and those who didn't, with no significant findings.
 A: In multiple impuation using for example mice, missing variables are first tentatively filled, which makes them suitable as predictor or even as response, and then they are iteratively imputed.
Check this R Code :
library(mice)
data("airquality")
airquality[5,1:2] # All data points are missing 
  Ozone Solar.R
5    NA      NA
# Impute using mice with just one impuation
imp <- mice(airquality[,1:2], method="norm.predict", m=1, maxit=3,seed=1)
complete(imp)[5,]` # Check the fifth observation
    Ozone  Solar.R
5 42.69252 186.4568

See my question Missing data - Regression imputation
Regrading MCAR test, I do not think you need to make this test at all , since MI assumes MAR mechanism, See also Does Little's MCAR test make sense?
A: It's complicated given numbers (or percentages) to accept or reject variables with missing data, because working with missing data is complicated and you can easily introduce bias.
From this blog I found something useful

Although imputation can improve the quality of the final data, care must be taken to choose an appropriate imputation methodology. Some methods of imputation do not preserve the relationship between variables. In fact, some can actually distort the underlying distributions.

While some could recommend you to reject the variable if you have more than 60% of missing values this isn't a rule and the percentages are inherent of your data and the study.
Maybe you want to look at the distribution of your variable, or the variance before the imputation. And something important is that you should consider what is the study that you are doing, and if this variables (with missing values) are needed for your model. 
