Dionysus (known by the Romans as Bacchus) was a Greek god who, among other things, gave and revoked Midas' power to turn everything he touched into gold. The original source of the list of comparisons comes from the book "The Jesus Mysteries" by Freke and Gandy. Let's check out this list (plus a few other claims people have added):
1. Dionysus was born of a virgin on December 25th and, as the Holy Child, was placed in a manger.
Actually, his birth was always celebrated on January 6th. Also, his mother, Semele, was impregnanted sexually by Zeus. He was never referred to as the "Holy Child" or placed in a manger in any version of the story. See Diodorus Siculus: Library Of History: Zeus And Semele
2. His birth was announced with a heavenly display and celestial music.
I can find no reference to either, and there is no "celestial music" in the Jesus story.
3. He was a traveling teacher who performed miracles.
This is true. However, this phrase loses any similarities with Jesus when we deal with the specifics of what Dionysus did. Jesus traveled in a limited area, while Dionysus supposedly traveled to most of the known world (including Greece, Persia and Arabia). Jesus' miracles were healings and such - all positive miracles. Dionysus' miracles were judgments against those who defied him.
4. He "rode in a triumphal procession on an ass" and "is often pictured astride a donkey, which carries him to meet his passion" a scene re-enacted with crowds "shout[ing] the praises of Dionysus and wav[ing] bundles of branches."
This claim mixes two things, one semi-valid, one invalid. Dionysus was dipicted riding a donkey while a crowd waved ivy branches - the typical homecoming for any royal figure. The crowd welcoming Jesus to Jerusalem were imitating this sort of homecoming, though using the traditional palm branches of Israel. So while this could be called a sort of imitation, it's an imitation committed by the people in the story itself, not by any writer, and had nothing to do with Dionysus in particular. The latter quotes come from the book "The Jesus Mysteries" by Freke and Gandy. Their only reference is to a depiction of a scene from Orphic eschatology which, oddly, has nothing to do with Dionysus.
5. He was a sacred king killed and eaten in a eucharistic ritual for fecundity and purification.
There exists an unofficial story (that is, not part of the general understanding of the Dionysus story) in which he is, as an infant, attacked by Titans who eat everything but his heart. Zeus destroys the Titans, and restores Dionysus from the remaining heart. Who would call the Jesus story a 'copycat' of that story? Taking this 'similarity' apart, yes, Dionysus was killed. His actual body was eaten, but since Jesus' body was not (the eating of Jesus' body is a metaphorical thing), this is not a comparison. Also, Dionysus wasn't eaten in any sort of ritual for fecundity or purification. In fact, the eating of Dionysus is clearly a bad thing (unlike the eating of Jesus' body) and is punished by death. Also, he wasn't a sacred king. The king was Zeus, not Dionysus.
6. Dionysus rose from the dead on March 25th.
Nowhere is the date of March 25th given in any Dionysus story. The date of his "resurrection" after his murder by the Titans is given as November 8th (and as shown in the above answer, this story is hardly similar to the story of Jesus' resurrection and is an unofficial story anyways). There is an ancient reference to Dionysus being "a god who renews himself and returns every year rejuvenated", but this doesn't involve death. Besides that, Jesus didn't rise from the dead on March 25th either. While an exact date is not given, most scholars believe that His crucifixion happened no earlier than March 28th, making His resurrection no earlier than March 30th.
7. He was the God of the Vine, and turned water into wine.
Dionysus was indeed "the God of the Vine". However, Jesus wasn't.
The earliest possible reference to Dionysus turning water into wine was by Achilles Tatius in the Greek Romance, "The Adventures of Leucippe and Clitophon" which was written in the 2nd century A.D. It mentions a Tyranian myth about Dionysus introducing wine to the world, with Dionysus calling it "the water of summer" and saying "This is the water, this is the spring". It's not clear whether this a real Tyranian myth being mentioned here (in which case it may be pre-Christian) or just something Tatius was inventing for the purposes of this story. Either way, Dionysus is not actually turning water into wine, but simply calling the wine a type of water. And we cannot reliably date this myth to any earlier than the second century A.D.
8. He was called "King of Kings" and "God of Gods."
Nope. These would be odd titles to give Dionysus, his being the son of Zeus, who is the main God in the Greek religion. The titles would only fit Zeus himself, and even he was never referred to by either of these titles.
9. He was considered the "only Begotten Son," "Savior," "Redeemer," "Sin Bearer," "Anointed One," and the "Alpha and Omega", and "Lord God of God born"
Of these, Dionysus is only referred to as 'savior'. And in the context in which he is referred to 'savior', he is saving people from the wrath of Pentheus, not from sin or eternal damnation. So even this is hardly a comparison to Jesus.
10. He was identified with the Ram or Lamb.
In one version, he is born with horns on his head like that of a ram. That's the only mention of a ram in any Dionysus literature, and doesn't compare to Jesus' story at all.
11. His sacrificial title of "Dendrites" or "Young Man of the Tree" intimates he was hung on a tree or crucified.
This was no a 'sacrificial' title in any sense. He was simply called 'Young Man of the Tree'. How does that suggest he was hung on a tree or crucified?
12. At his trial, Dionysus is described by Freke and Gandy as "a quiet stranger with long hair and a beard who brings a new religion."
Dionysus was hardly quiet before King Pentheus, but engaged in quite a bit of dialogue with him. As for "long hair and a beard", most men wore both in those days. In fact, the Bible says nothing about Jesus having long hair or a beard. We only assume He did because they were common for middle-eastern Jews in those days.
13. He offers his followers the chance to be born again through the rites of baptism.
Nope. The followers of Dionysus never claimed to be 'born again' and their 'baptism' had to do with waving a fan above their heads, not submersing them in water.
14. His followers await his return as the judge during the Last Days.
This is not true of any version of the Dionysus story.
Apollodorus on Dionysus (2nd century B.C.)
Diodorus Siculus on Dionysus (3rd century B.C.)
1st Homeric Hymn to Dionysus (6th or 7th century B.C.)
2nd Homeric Hymn to Dionysus (6th or 7th century B.C.)
Ovid, "Pentheus And Bacchus" (late 1st century B.C. or early 1st century A.D.)