#!/usr/bin/perl
use strict;
use warnings;
# cf pl_test/751890.pl

my @myarray=qw/two three threee fouuuuur five septigesimal sodium hexa
+flouride/;
my @vowels=qw/a e i o u/;
my @words_having_3_vowels;

for my $var(@myarray) {
    my $seen;
    for my $vowel(@vowels) {
        if ($var =~ m/$vowel/) {
            $seen++;
            if ($seen == 3) {
                push @words_having_3_vowels, $var;
                next;
            }
        }
    }
}

for my $words(@words_having_3_vowels) {
    print "$words \n";
}

=head   execution output:
pl_test/751890_a.pl
septigesimal
sodium
hexaflouride

=cut
[download]

Now, echoing responses above, please read about <code>...</code> (or <c>...</c> tags in Writeup Formatting Tips or Markup in the Monastery so that your code is readable (what you did is, but more complex code, or that using characters like [ would NOT be readable without code tags.

And read also How do I post a question effectively? because "But not getting the expected results." is not a description of the problem which clarifies your problem. Also, as noted above, your description of what you're trying to achieve is ambiguous./p>

Finally, the original code, cleaned up to use strict; and use warnings;; with variables declared (albeit, not constrained as to scope with appropriate indentation (well, one legible version of 'appropriate' indentation, of which there are many);

#!/usr/bin/perl
use strict;
use warnings;

my @myarray=qw/two three threee fouuuuur five septigesimal/;
my @vowels=qw/a e i o u/;
my @words_having_3_vowels;
foreach(@myarray) {
    my $var=$_;
    for ( my $p=0; $p<5 ; $p++ ) {
        for ( my $q=0; $q<5 ; $q++ ) {
            for ( my $r=0; $r<5 ; $r++ ) {
                if ( $var =~ /^(.*)($vowels[$p])(.*)$vowels[$q](.*)$vo
+wels[$r](.*)$/ ) {
                    push @words_having_3_vowels, $var;
                }
            }
        }
    }
}
for my $result(@words_having_3_vowels) {
    print "|$result|\n"; 
}

=head  execution output:

|threee|
|fouuuuur|
|fouuuuur|
|septigesimal|
|septigesimal|
|septigesimal|
|septigesimal|
|septigesimal|
|septigesimal|
|septigesimal|
|septigesimal|

=cut
[download]

This is unnecessarily hard on you and the future reader and could be avoided by <insert> either of</insert> the technique ikegami cites; it's also far more work to type (and track) than linuxer's solution. Note, however, that his output is a per-word vowel count; not the words with 3 or more vowels, and also that lines 20-22 in my "cleaned up" version change your print command (line 16 by my count) to a print-in-a-loop that prints a word (wrong: each time its vowel-count exceeds two) (correction: one or more times for reasons I'm too lazy to explore and elucidate).

Update Inserted the italicized clarification re ikegami's suggestions.
update 2 Original statement re output, now stricken, is incorrect.