Use seqtk (LINK).

seqtk subseq in.fa reg.bed > out.fa

You can use the following code:

cat id_file.txt | while read id start stop; do echo ">"$id >> output_file.fasta ;  perl -ne 'if(/^>(\S+)/){$c=grep{/^$1$/}qw('$id')}print if $c' seq-file.fasta | tail -n +2 | cut -c $start-$stop >> output_file.fasta ; done

To use this code you first need to delete empty lines from your sequence file:

perl -p -i -e "s/^\n$//g" seq-file.fasta

if you don't want to modify the original file, try:

perl -p -e "s/^\n$//g" seq-file.fasta > new_seq-file.fasta

If each of your sequences is one line, you can use this code too:

cat id_file.txt | while read id start stop; do echo ">"$id >> output_file.fasta ;  grep -A 1 ">$id" seq-file.fasta  | cut -c $start-$stop >> output_file.fasta ; done

If you want to add start and stop to the header, you can use this echo instead of echo ">"$id:

echo ">"$id"_"$start"_"$stop

The output:

>seq1_5_15
AGAGCCTTGTC
>seq2_2_10
ACTCAGTTT
>seq3_10_20
CCGTGGAGGAG

You have tagged R, but are asking for any tools. Can you clarify your requirements?