I saw in the book of Rubin (1987) that an increase in variance of estimation will occur due to nonresponses. But I wonder the reason behind this. Thanks for your shares!
The "increase" is relative to a fully responding sample. So if you sample $n $ but only $ n_o< n$ units actually respond, then the variance will be higher - just like if you sampled less people.
There may also be systematic differences between those who respond and those who don't respond - this is non-response bias. This may further inflate your variance (technically its the mean square error that is inflated due to bias).
I would say that possible bias arising out of differences in distribution of survey variables amongst respondents and non-respondents is much more dangerous situation than increase in the sampling variation.
In any case there must be some method to adjust inclusion probabilities from which you calculate weights to transform sample level statistics into population ones.