Computing the Levenshtein distance quickly Given a huge database of allowed words (alphabetically sorted) and a word, find the word from the database that is closest to the given word in terms of Levenshtein distance.

The naive approach is, of course, to simply compute the levenshtein distance between the given word and all the words in the dictionary (we can do a binary search in the database before actually computing the distances).

I wonder if there is a more efficient solution to this problem. Maybe some heuristic that lets us reduce the number of words to search, or optimizations to the levenshtein distance algorithm.

Links to papers on the subject welcome.
What you are asking about is the problem of near-neighbor search under the edit distance. You didn't mention whether you're interested in theoretical results or heuristics, so I'll answer the former. 

The edit distance is somewhat nasty to deal with for building near-neighbor search structures. The main problem is that as a metric, it behaves (sort of) like other well known bad metrics like $\ell_1$ for the purpose of dimensionality reduction and approximation. There's a rather vast body of work to read on this topic, and your best source is the set of papers by Alex Andoni: by following pointers backward (for example from his FOCS 2010 paper) you'll get a good set of sources.