The two approaches offered thus far fail with large data sets as (amongst other memory issues) they create is.na(df), which will be an object the same size as df.
Here are two approaches that are more memory and time efficient
An approach using Filter
Filter(function(x)!all(is.na(x)), df)
and an approach using data.table (for general time and memory efficiency)
library(data.table)
DT <- as.data.table(df)
DT[,which(unlist(lapply(DT, function(x)!all(is.na(x))))),with=F]
examples using large data (30 columns, 1e6 rows)
big_data <- replicate(10, data.frame(rep(NA, 1e6), sample(c(1:8,NA),1e6,T), sample(250,1e6,T)),simplify=F)
bd <- do.call(data.frame,big_data)
names(bd) <- paste0('X',seq_len(30))
DT <- as.data.table(bd)
system.time({df1 <- bd[,colSums(is.na(bd) < nrow(bd))]})
# error -- can't allocate vector of size ...
system.time({df2 <- bd[, !apply(is.na(bd), 2, all)]})
# error -- can't allocate vector of size ...
system.time({df3 <- Filter(function(x)!all(is.na(x)), bd)})
## user system elapsed
## 0.26 0.03 0.29
system.time({DT1 <- DT[,which(unlist(lapply(DT, function(x)!all(is.na(x))))),with=F]})
## user system elapsed
## 0.14 0.03 0.18