Delete line from text file
What you can do is:


opening a file (to read)
opening a new file (to write)
since a line can be long (more than arduino's memory), it's best to read it character by character, thus:
use a boolean to denote when to start removing a line (set to e.g. Remove = TRUE)
For every character you read:


If Remove is TRUE, don't write to the output file, otherwise write it
If the character is \n or \r (check how line ends are), set the boolean to False (e.g. continue writing)

Remove file to read
Rename written file