Searching the docs for "timeout" found a solution.

# Check for new File::Tail object every $timeout seconds.
my $timeout = 60;

for (;;) {
    my (undef, undef, @pending) =
        File::Tail::select(undef, undef, undef, $timeout, $file);

    for (@pending) {
        my $line = $_->read();
        print $line;
    }
}
[download]

Update: You could even simplify your signal handler (always a good thing) and your code in general as follows:

use File::Tail;

# Check for new File::Tail object every $timeout seconds.
my $timeout = 60;

my $qfn     = $ARGV[0];
my $new     = 1;
my $changed = 0;
my $tail;

$SIG{INT} = sub { $changed = 1 };

for (;;) {
    if ($changed) {
        $changed = 0;
        open(my $fh, '<', '/tmp/filename')
            or die("Cannot open file: $!\n");
        chomp( $qfn = <$fh> );
        $new = 1;
    }

    if ($new) {
        $new = 0;
        $tail = File::Tail->new(
            name        => $qfn,
            maxinterval => 3,
            adjustafter => 2,
            tail        => -1,
        );
    }
    
    my (undef, undef, @pending) =
        File::Tail::select(undef, undef, undef, $timeout, $tail);

    for (@pending) {
        my $line = $_->read();
        print $line;
    }
}
[download]