Since you usually assume missing at random (MAR) conditional on the observed data when doing multiple imputation, it is important to include all the data that may give information on the unobserved missing value or the reason for why it is missing in the multiple imputation. If key variables are ommitted, then the assumptions behind the imputations are likely to be violated.
Deciding on the variables that should be included based on some logistic regression for the missingness indicator (presumably that was the idea of what was mentioned?) is almost certainly not a good idea for a number of reasons: Firstly, this would say nothing about how much information on the missing values a variable contains. Secondly, if there are few missing values, no or hardly any variables would be associated with a low p-values despite possibly being extremely important. Thirdly, almost always when you look at strategies that involve doing some kind of hypothesis tests to decide how to conduct an analysis, these strategies tend to have poor properties (I'm generalizing, but I'd be interested to hear of any case where such an approach has been shown to have good properties).