Someone already pointed out that you need to define your dispatch table a little differently. Your example is creating a hash of scalar references, instead of a hash of subrefs. I won't belabor that point since it's already been made.

I did want to mention that your dispatch table needn't use named subs at all. Consider the following code:

my %dispatch = ( 
    DoThis => sub { print "You asked me to do this.\n"; },
    DoThat => sub { print "You asked me to do that.\n"; }
);
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For trivial subs, defining them inline like this can be a simple, convenient, and readable solution.

Another solution is the switch construct:

SWITCH: for ( @commands ) {
    m/^This/ && do { print "You asked me to do this.\n";
                     next SWITCH;
                   };
    m/^That/ && do { print "You asked me to do that.\n";
                     next SWITCH;
                   };
    m/^Another/ && Another_Way();
    m/^Quit/ && do { print "Goodbye.\n";
                     last SWITCH;
}

sub Another_Way {
    print "You asked me to do it another way.\n";
}
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Of course there are other ways to do it as well... This is Perl after all.