Do these sentences have the same meaning?

No the do not have the same meaning.

The first sentence is telling someone you really ‘want’ to do something but the second sentence tells someone that doing it would be interesting but you are not over concerned whether yo do it or not.

“Desire” has a much stronger connotation than “very interesting.”

There are many reasons why you want to do something, even if it is not particularly interesting. For example, doing something may save you money.

Way different.

The first is an indication of a possibility in the future. The second is an activity occurring now.

Consider this:

I desire to kiss that woman.

The way she uses her tongue while kissing is very interesting to me.

Although you were asking only about meaning, the sentence structure of each sentence is awkward and not correct in standard English. “Desire” is usually followed by a direct object, not an infinitive and the second sentence would read better as, “I am very interested in doing this.” Active rather than passive voice.

No, not exactly.

They do not have the same meaning. Both are grammatically correct: “to do” and “doing” are forms of the verb—infinitive and participle, respectively—being used as nouns. We do this all the time: I like to read, I like reading.

And both expressions as written trigger a sense that “we just wouldn’t say it that way,” meaning that they’re not what we generally see as idiomatic American English. We typically don’t use “desire” as a verb much any more. And we generally say “I’m interested in” rather than “It is interesting to me.” But neither form is incorrect.

If I were in an old British movie, I might well tell the hotel desk clerk, “I desire a room.” And if I wanted to keep the object of my curiosity a bit at arm’s length, I might well say “Watching a Venus flytrap process its food is interesting to me,” but not “I’m interested in watching carnivorous plants devour insects.”

But the real difference is in the word choices themselves: “desire” is a much stronger expression than “interest,” although they might be seen on the same continuum. One’s interest might never reach the peak of desire, and one’s desire might never weaken to the level of mere interest. They might also be so far separated in shades of meaning that you would never use one in place of the other; that depends on context.

So it comes down to meaning, not structure.

Neither sentence is in the passive voice.

No. Desire and interest are not synonyms.

@Jeruba Yes, that is what I said in my first response as have others. I will leave the grammar to you in future.

I must be a dinosaur. I can easily hear myself saying that I desire to do something.

@Hawaii_Jake, I qualified my generalities. That use is by no means obsolete, so you’re not yet fossilized. We are more apt to use it as a noun, though (“I have a lifelong desire to visit Paris in winter”) or in the negative (“I have no desire to interrupt your activities.”) Here is one use that is pretty much obsolete: a line of dialogue from G&S’s H.M.S. Pinafore (1878), putting the verb in the imperative rather than the indicative mood:

CAPT. I am the last person to insult a British sailor, Sir Joseph.
S IR JOSEPH. You are the last person who did, Captain Corcoran. Desire that splendid seaman to step forward.

They are not the same

Those two sentences are not the same