What is the value of a dime on which the letters 'm' and 'e' are missing on the word 'dime'?

Probably ten cents. Missing letters are hard to validate. If it left the mint that way, I'd say it would be worth a lot, but being able to prove it may be difficult. Anyone can remove a couple of letters. On the other hand, an expert in error coins should be able to determine if the error was due to a filled die or other minting error. If it doesn't look like it was damaged, I'd take it to a dealer for an evaluation. They may laugh and toss it back to you or they might make an offer. If they do make an offer, take the coin and visit a couple of other dealers!