I would use dplyr to remove the earliest date for each group. I'm providing some data here.
library(dplyr)
df <- structure(list(ID = c(1, 1, 1, 2, 2, 2), time = structure(c(1325485800,
1325487600, 1325489400, 1325491200, 1325493000, 1325494800), class = c("POSIXct",
"POSIXt"), tzone = "")), class = "data.frame", row.names = c(NA,
-6L))
df.updated <- df %>%
dplyr::group_by(ID) %>%
dplyr::slice(-which.min(time))
Be sure to provide data when asking a question to give a good reproducible example. You can do this through dput(head(df)) to provide some of your data (as usually it only takes a little data to solve an issue).