Yes.

Oh wait, did you want to know what the problem is?

Read the DBI docs. You'll find there that the method do() is used on a database handle - in this case, that's your $dbh. Thus you need to do this:

my $sql = "INSERT into 1985_onwards_slim_I
  (Year_of_death, Ageyears, Sex, Undcause, OAcode)
    SELECT Col003, Col020, Col023, Col029, Col051
        from 1985_onwards_full_data_set";
$dbh->do($sql);
[download]

If you want to prepare and execute, the sequence is like this:

my $sth = $dbh->prepare($sql);
$sth->execute();
[download]

Also, I recommend some Error-checking - always a plus.