Before attempting to remove metal from a reed, you need to support the reed so you don't push it through the slot. A thin shim like a .003 spark plug gapper works well, as does a razor blade. You can even use a business card--anything small and thin will do.
Be careful not to push the reed sideways in its slot, or the reed won't vibrate freely. Also, be careful about filing the reed edges, which can cause burrs that catch on the slot-edges as the reed vibrates through the slot. If you get a burr you can shave or file it off.
You need to use a chromatic tuner to check the pitch of the note. You may notice small pitch differences between a note played with the cover off and when the cover is in place. As you get experience doing it you'll be able to judge how to read your tuner and end up with the right note.
Harmonica reeds often go flat, and sometimes you can tune the reed back up to pitch. However, if the reed has gone flat by a semitone or more, it is probably fractured, and tuning the reed may not work. In fact, it may stress the reed enough to cause it to break--but don't worry, it was broken already.
Many harp players do not tune their harps often enough. With a little practice you'll know just how much metal to remove, and where, and it won't take long at all to get your harp all tuned up.
Caution: be sure to check the tuning with a chromatic tuner first to see what the reference frequency is. Historically, the frequency of an A note is used as the reference frequency, but not everywhere uses the same frequency! A=440 cycles per second is very common, but harps are often tuned to A=441 or 442 or even higher, because harps are often played slightly flat, so tuning them sharp makes the resultant note fit better with other instruments.
If you tune each note exactly to pitch according to your tuner, the result will be in so-called equal temperament. Equal temperament is common on many models of harps, such as the Lee Oskar Major Diatonic and the Hohner Golden Melody. This tuning is optimized for playing single notes and melodies, but the chords will sound a bit out. To make certain chords sound better, many harps are tuned to a justified (or just) intonation. Just intonation involves modifying the pitch of certain notes to make some chords sound better--but melody notes may sound flat or off key. Various compromised intonations that aren't quite just intonation and aren't equal temperament have been devised to try to work as well as possible for both melody notes and chords.
For an extensive discussion of tunings and temperaments, see Pat Missin's
web page at:
http://www.users.globalnet.co.uk/~patm.
In addition to keeping your harp in tune, various special tunings can be done to provide different notes (without requiring special bending or overbending techniques) and different chords. Examples include the Natural Minor, country tuning, and Lee Oskar's Melody Maker tuning. Using the above procedures, it is relatively easy to build your own specially tuned harps. Pat Missin's web page contains his "Altered States" document, which contains hundred of different tunings for both the diatonic and chromatic harps.
The simplest special tuning is the "country" (or in Huang terminology, "jazz") tuning, which has the 5 draw note raise a half step. This allows the major 7 note to be played without requiring an overblow, and also yields a Major 7 chord. The normal dominant 7 note is still available by using an ordinary draw bend. The extra note provided by the country tuning is often required for standard's melodies.
Special tunings are great for special songs, and can really help you break out of ruts by providing new sounds to old playing patterns.