Option 1

Use the fact that a data.frame is a list of columns, then use do.call to recreate a data.frame.

do.call(data.frame,lapply(DT, function(x) replace(x, is.infinite(x),NA)))

Option 2 -- data.table

You could use data.table and set. This avoids some internal copying.

DT <- data.table(dat)
invisible(lapply(names(DT),function(.name) set(DT, which(is.infinite(DT[[.name]])), j = .name,value =NA)))

Or using column numbers (possibly faster if there are a lot of columns):

for (j in 1:ncol(DT)) set(DT, which(is.infinite(DT[[j]])), j, NA)

Timings

# some `big(ish)` data
dat <- data.frame(a = rep(c(1,Inf), 1e6), b = rep(c(Inf,2), 1e6), 
                  c = rep(c('a','b'),1e6),d = rep(c(1,Inf), 1e6),  
                  e = rep(c(Inf,2), 1e6))
# create data.table
library(data.table)
DT <- data.table(dat)

# replace (@mnel)
system.time(na_dat <- do.call(data.frame,lapply(dat, function(x) replace(x, is.infinite(x),NA))))
## user  system elapsed 
#  0.52    0.01    0.53 

# is.na (@dwin)
system.time(is.na(dat) <- sapply(dat, is.infinite))
# user  system elapsed 
# 32.96    0.07   33.12 

# modified is.na
system.time(is.na(dat) <- do.call(cbind,lapply(dat, is.infinite)))
#  user  system elapsed 
# 1.22    0.38    1.60 


# data.table (@mnel)
system.time(invisible(lapply(names(DT),function(.name) set(DT, which(is.infinite(DT[[.name]])), j = .name,value =NA))))
# user  system elapsed 
# 0.29    0.02    0.31

data.table is the quickest. Using sapply slows things down noticeably.