I'm not entirely sure I understand the rules by which you want newlines to be added to the output or not. In the following, if you don't want newlines in the output, change the two occurrences of "\n" to " ".

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

my %data;
my $cur_tag;
while (<DATA>) {
    chomp;
    if ( my ($tag,$line) = /^(X\S+)\s+(.+)$/ ) {
        $data{$tag} .= "\n" if exists $data{$tag};
        $data{$tag} .= $line;
        $cur_tag = $tag;
    }
    elsif ($cur_tag && !/^BEGIN_TAG/) {
        $data{$cur_tag} .= "\n".$_;
    }
}

for my $tag (sort keys %data) {
    print "$tag: $data{$tag}\n";
}

__DATA__
BEGIN_TAG
X1 test1 test1 test1
X2 no1 no1 no1 no1
X3 yes1 yes1 yes1
BEGIN_TAG
X1 test2 test2 test2
test3 test3 test3
X2 no2 no2 no2 no2 no2
no3 no3 no no3 no3
no4 no4 no4 no4 no4 no4 no4
X3 hi2 hi2 hi2 hi2 hi2 hi2
hi3 hi3 hi3 hi3
hi4 hi4
[download]

Output:

X1: test1 test1 test1
test2 test2 test2
test3 test3 test3
X2: no1 no1 no1 no1
no2 no2 no2 no2 no2
no3 no3 no no3 no3
no4 no4 no4 no4 no4 no4 no4
X3: yes1 yes1 yes1
hi2 hi2 hi2 hi2 hi2 hi2
hi3 hi3 hi3 hi3
hi4 hi4
[download]

Update: Inverted "\n" vs " ".