Make it possible to set up your program with a different database handle. In your test program, open a handle to a file or memory database with just enough test data to make things work.

I used DBD::CSV in a project, stuffed the applicable data after __DATA__, and wrote a small sub to dump things to a CSV file. Very simple:

sub make_db {
    open(OUT, '>page_data') or die "Can't open output db: $!";
    while (<DATA>) {
        chomp;
        print OUT $_, "\015\012";
    }
    close OUT;
}
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Depending on how tricky your SQL calls are, you may have to get more tricky here. Hmm, it would be good to have someone write a DBI testing module. Any takers?