use warnings;
use strict;
use JSON;

 . . .

# Build a JSON parser
my $json = JSON->new->allow_nonref;

# Process the lines in the file, we're labelling the WHILE loop to make it obvious
# where we skip to after handling the exception
JSON_RECORD:
while (my $line = <INFILE>) {
    # We'll declare the variable so we can use it outside of the eval
    my $data;

    # Parse the data inside the eval so we can catch the error
    eval {
        $data = $json->decode($line);
    };

    # Now we can check $@ to see if we had an exception
    if ($@) {
        # parse failed, so log it and let the user know
        print "JSON parse failed on line $.\n";
        print "error was: $@\n";
        print "JSON we tried to parse <<$line>>\n\n";
        # Now that we've handled the exception, skip to the next recor
        next JSON_RECORD;
    }

    # OK if we get here, we've successfully parse the JSON data, so we can do what
    # we normally do now.

    . . .
}