In that case, set the $flag-variable to 0 when you expect to read a 'gene'-line, and set it to 1 when you expect to read a 'CDS' or 'exon'-line.
You will need to keep the @BMB-variable declared before going into the while-loop, since you need to keep the data you put in there (when $flag was 0) to be still there when you're going through the while-loop with $flag being 1.
Note that the error-handling is not present in this version, but you can add that yourself.
This results in the following:

#!/usr/bin/perl

# Task: Extract GeneID-Number and gene information

use strict;
use warnings;

my $in;
my $data;
my @array;
my $array;
my $GeneID;
my @BMB;
my $flag = 0;
my %hash;
my $hash;

# 1) open the .gff Inputfile and while reading line by line split $dat
+a at each tab and put them in the @array

open $in, '<', "Genomteil.gff" or die $!;
while ($data = <$in>) {
    @array = split (/\t/, $data);
    if ($flag == 0) {
        if ($array[2] =~ /gene/) {    #if you find the word 'gene' a t
+extbloxk follows which contains some information I want to extract an
+d put in an array)
            $flag = 1; # Set the flag. We will be expecting a 'CDS' or
+ 'exon'-line next
            @BMB = ($array[3], $array[4], $array[6]);    #the array wi
+ll be used as values for my hash later
        } ## end if ($array[2] =~ /gene/)
        if ($array[8] =~ /.*;db_xref=GeneID:(\d+)\n/) {    #if you fin
+d the word 'GeneID' extract the following number and put it in my has
+h (as key), then put the array in my hash
            $GeneID = $1;
        } ## end if ($array[8] =~ /.*;db_xref=GeneID:(\d+)\n/)
    } elsif ($flag == 1) {
        if ($array[2] =~ /CDS/) {
            push (@BMB, $array[2]);
            #put more data in my array
        } elsif ($array[2] =~ /exon/) {
            push (@BMB, $array[2]);
        }
        @{$hash{$GeneID}} = @BMB;
        $flag = 0; # Reset the flag. We will be expecting a 'gene'-lin
+e next
    }
} ## end while ($data = <$in>)

close $in;

my $BMB;
while (($GeneID, $BMB) = each %hash) {
    print "$GeneID => $BMB[0]\n";
}
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