I checked the rest of the program for some strange interaction with that file and filehandle name

That's why you don't use bareword filehandles.

use strict;
use warnings;
use 5.010;


#Create some text files:

my @fnames = ('test1.txt', 'test2.txt');

for my $fname(@fnames) {
    open my $OUT, '>', $fname
        or die "Couldn't open $fname: $!";

    for my $num (1 .. 10) {
        say $OUT "$fname -- $num";
    }
}

#------------------

sub print_error {
    say 'error';
}

#------------------

#Read the text files:

FOR: 
for my $fname('test1.txt', 'test2.txt') {
    open my $INPUT, '<', $fname 
        or print_error();

    while(my $line = <$INPUT>) {
        print $line;
    }

    print "\n";
    
}

--output:--
test1.txt -- 1
test1.txt -- 2
test1.txt -- 3
test1.txt -- 4
test1.txt -- 5
test1.txt -- 6
test1.txt -- 7
test1.txt -- 8
test1.txt -- 9
test1.txt -- 10

test2.txt -- 1
test2.txt -- 2
test2.txt -- 3
test2.txt -- 4
test2.txt -- 5
test2.txt -- 6
test2.txt -- 7
test2.txt -- 8
test2.txt -- 9
test2.txt -- 10
[download]

When the input line operator, <>, reads end-of-file, it returns a false value that causes the while loop to end.

How about using the 3-arg form of open()?

open(DB,"<$F")

#compare to:

open my $INFILE, '<', $fname
  or die "Couldn't open $fname: $!"
[download]

How about printing out the filename in the loop?

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