sed is your friend.

sed '2~4s/^\(.\{4\}\)//' myfile > newfile

Translated that says: starting from line 2, substitute the first 4 characters every 4th line with nothing (i.e. remove them).

If you don't want to write to newfile, run:

sed -i '2~4s/^\(.\{4\}\)//' myfile

to edit myfile "in place".

EDIT

I think you will also want to remove the corresponding characters from the quality score lines. So you should run:

sed '2~2s/^\(.\{4\}\)//' myfile > newfile

Other useful command line tools for text processing: grep, awk, cut, paste, head, tail.

As an alternative approach - the FASTX-Toolkit provides a number of command line utilities for manipulating FASTQ and FASTA sequence files.

fastx_trimmer has lots of options for trimming sequences in a variety of ways. To achieve what you are after (trim first 4 bases from each read):

fastx_trimmer -f 5 -z -i infile.fq.gz -o outfile.fq.gz

The -f 5 option is the key one here, this says that the 5th nucleotide of each sequence is the first one you want to keep (i.e. the first 4 are discarded).

Perl-one-liner, trimming read sequence and qualities lines:

cat file.fq | perl -plane '$ln++; s/^....// if ($ln % 2 == 0)' > trimmed.fq