Slurp the file into a scalar and then do a single match on it per value you're looking for. Like so:

local $/;
my $file = <IN>;
close IN;

@array=('file', 'this', 'dog', 'forward');

foreach $searchstring(@array) {
  if ($file=~m/$searchstring/i) {
    push @stringsfound,$searchstring;
  }
}
[download]

The foreach block can also be written as

@stringsfound=grep {$file=~/$_/i} @array;
[download]

I changed your regular expression, omitting the (.*)? since it didn't seem to be fulfilling any useful function (ordinarily this would capture any characters before $searchstring into $1, but you're not using $1 for anything).