Is It Safe to Have my Two Front Teeth Filed Down? (photo)

If the adjustments are little more than aggressive polishing, taking "just a little" enamel, you should be fine.  But, you should be aware that if more aggressive adjusting is done, the teeth may need to be restored (crowns or veneers).

Shaving teeth enamel is absolutely safe as long as it is done within reason.
In a healthy individual average thickness of outside layer of the tooth, which is the hardest substance in the body, is about 1mm. It can safely be shaved up to 0.5mm and still preserve tooth integrity.

There is a procedure that dentists can do  called "cosmetic contouring".  When small changes to the shape of a tooth can make a big difference to the appearance of a your teeth this is a good solution.  The key words here are "small changes".  As long as you are just 'tweeking' minor issues so that your front teeth will blend more harmoniously with the teeth around them it should be OK.  Be sure that your dentist is experienced in esthetic dental procedures.

Your tooth is all you have and will not grow again.  This may seem obvious but take great care in asking a dentist to file your teeth down.  What will it look like afterwards? Do a mockup with marking a dark pencil masking the edges that may be adjusted. take pictures.  Your teeth and smile and your face all come into play.  You have a high smile line with what can commonly be referred to as a rather "gummy smile".  This can be altered easily enough (and reversibly) and suddenly your teeth look OK without filing them down? An alternative is to make the adjacent teeth a little longer (bonding composite) again without harming them and reversible (remove composite if not pleased with result). This can be done as a mockup before  commencing any irreversible tooth destruction.  Your orthodontic result is great now you need to look after your teeth as best as possible.  Veneers/crowns all destroy precious tooth enamel and will need replacement later so don't demand this approach unless all other alternatives are tried and exhausted.  Talk to your referring dentist they should be able to help with a minimum of harm to any teeth.  I don't understand how you are supposed to bite and why you cannot bite like you are supposed to but one picture will not make a diagnosis I am afraid.