In your sample data there are no NAs, remember this stands for "Not Available" and basically means there is no value at all, so you can't compare something to nothing, the correct way would be to use is.na() function, see this example with some NAs added to your sample data.
library(tidyverse)
psppipla <- c("Not at all", "Very little", "Some", "A lot", "A great deal", "No answer")
vote <- c("Yes", "No", NA, "Refusal", NA, "No answer")
data_Austria <-data.frame(psppipla, vote)
data_Austria %>%
filter(!is.na(vote)) %>%
ggplot() +
geom_bar(mapping = aes(x=psppipla, fill=vote))

Also, have in mind that dplyr functions like filter() do not perform in-place modifications, they return a new data frame instead, if you want changes to persist, you have to explicitly assign them to a variable (it could be the same one if you want to overwrite it).