"The dirges of his hope" in this poem means "the sad songs sung to express that hope itself has died" or, more simply, "the sad sounds made by a hopeless man." Let's explore that idea.

Dirges are sad songs sung at funerals or in honor of the dead. They express grief as well as respect for the person who has died.

In "The Raven," after the bird flies into the speaker's apartment and says "Nevermore" two times, the speaker realizes that "nevermore" is probably the only word the bird knows. Reasoning that the bird was probably owned by someone who was very unhappy and often talked (or sang) about his burdens, the speaker reasons that the bird heard his owner say "nevermore" all the time, so that's the word the bird also began to say. Here's the stanza in which all of that happens:

Startled at the stillness broken by reply so aptly spoken,
"Doubtless," said I, "what it utters is its only stock and store,
Caught from some unhappy master whom unmerciful Disaster
Followed fast and followed faster till his songs one burden bore-
Till the dirges of his Hope that melancholy burden bore
Of 'Never- nevermore'."

As you can see, the phrase "Till the dirges of his Hope" doesn't make much sense by itself yet, so let's understand it in the context of a bigger phrase:

Disaster Followed...till his songs one burden bore—Till the dirges of his Hope that melancholy burden bore Of "Never—nevermore."

The sentence above means "Disaster kept happening to the man until he just expressed his burdens in a song made up of one sad word: 'nevermore.'" Or, more simply, "Bad things happened to the man until all he said all the time was 'Nevermore.'"

So, "the dirges of his hope" means "the funeral songs for his hope" or "the sad songs sung to express a total loss of hope."