Java 8, 7054 6467

This is inspired by (but not copied from) the builtin java.lang.String.hashCode function, so feel free to disallow according to rule #2.

w -> { return w.chars().reduce(53, (acc, c) -> Math.abs(acc * 79 + c)) % 16777216; };

To score:

import java.nio.file.Files;
import java.nio.file.Paths;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.function.Function;

public class TweetableHash {
    public static void main(String[] args) throws Exception {
        List<String> words = Files.readAllLines(Paths.get("british-english-huge.txt"));

        Function<String, Integer> hashFunc = w -> { return w.chars().reduce(53, (acc, c) -> Math.abs(acc * 79 + c)) % 16777216; };

        Map<Integer, Integer> hashes = new HashMap<>();
        for (String word : words) {
            int hash = hashFunc.apply(word);
            if (hash < 0 || hash >= 16777216) {
                throw new Exception("hash too long for word: " + word + " hash: " + hash);
            }

            Integer numOccurences = hashes.get(hash);
            if (numOccurences == null) {
                numOccurences = 0;
            }
            numOccurences++;

            hashes.put(hash, numOccurences);
        }

        int numCollisions = hashes.values().stream().filter(i -> i > 1).reduce(Integer::sum).get();
        System.out.println("num collisions: " + numCollisions);
    }
}