First capture the Data::Dumper output in a string, then prepend the timestamp:

my $message= Dumper $myvar;
$message=~ s!^!$timestamp !mg;
print $message;
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See the Data::Dumper $Data::Dumper::Pad configuration variable.

IMHO, it would make sense to encapsulate everything in a subroutine along the lines of

use strict;
use warnings;

sub tprint {
    my $msg = join $,//'' ,@_;
    my $tstamp = localtime()." ";
    $msg =~ s/^/$tstamp/gm;
    print $msg;
}

tprint "something\nelse\n";
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BUT to completely replicate the interface of the print function is beyond my Perl skills. I have no idea how one could now provide a file handle to print into file.