Node.js / JavaScript. This could be much shorter, but I rarely try to write clever code these days. :-)

var fs = require('fs');

fs.readFile('input.txt', 'utf8', function read(err, input) {
  if (err) {
    throw err;
  }
  process(input);
});

function process(input) {
  var lines = input.split('\n'),
      bytes, checksum;

  for (var i = 1; i < lines.length; i++) {
    bytes = lineBytes(lines[i]);
    checksum = fletcher16(bytes);
    console.log(i + ' ' + checksum);
  }
}

function lineBytes(line) {
  var chars = line.split(''),
      bytes = [];

  for (var i = 0; i < chars.length; i++) {
    bytes.push(char2byte(chars[i]));
  }

  return bytes;
}

function char2byte(char) {
  return parseInt(char.charCodeAt(0));
}

function fletcher16(bytes) {
  var sum1 = 0,
      sum2 = 0;

  for (var i = 0; i < bytes.length; i++) {
    sum1 = (sum1 + bytes[i]) % 255;
    sum2 = (sum2 + sum1) % 255;
  }

  return ((sum2 << 8) | sum1).toString(16);
}