Here's a quick & dirty regex method (note that it's destructive to $s, and that it wouldn't support additional tags, and the tag order matters, but hopefully it's a good start):

#!/usr/bin/perl
use strict;
undef $/;
my $s = <DATA>;

my @questions;
while( $s =~ s!
  (?:\\begin{instructions}(.*?)\\end{instructions}\s*)?
  \\begin{exer}\s*
  \\begin{exertext}(.*?)\\end{exertext}\s*
  \\begin{soln}(.*?)\\end{soln}\s*
  (?:\\begin{answer}(.*?)\\end{answer}\s*)?
  \\end{exer}
    !!sx ){
  push @questions, {
    instructions => $1,
    exertext => $2,
    soln => $3,
    answer => $4,
  };
}

use Data::Dumper;
print Data::Dumper::Dumper \@questions;
__DATA__
#### place the sample input here ####
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