Is "He is risen" Correct?

Given that this has appeared around Easter (albeit a couple of days early in my calendar), I'm going to answer on the basis of the phrase

Christ is risen! He is risen indeed, alleluia!

which is used in some traditions.

There are two ways of looking at this.

1. It's an archaic use of English which conjugates verbs of motion with be in present perfect, in much the same way as French still does. The statement is the equivalent of "Christ has risen" and is stating a present-perfect fact.

2. It's stating an eternal truth that not only did he rise all those years ago, but he remains risen now. That is, not just "he has risen" but "he is risen" and risen is closer to being an adjective than a pure participle.

Affirming Andrew Leach's answer, the Paschal Greeting can be classified as a set phrase in many languages--especially those influenced by Orthodox Christianity.

He is risen is perceived in modern English as a predicate adjective, but it is technically an archaic present perfect construction from Matthew 28:6:

He is not here: for he is risen, as he said. Come, see the place where the Lord lay.
KJV Bible Gateway
Emphasis mine

The translators of the KJV used this construction to translate the aorist passive ἠγέρθη in the Greek text:

οὐκ ἔστιν ὧδε, ἠγέρθη γὰρ καθὼς εἶπεν: δεῦτε ἴδετε τὸν τόπον ὅπου ἔκειτο:
qbible.com
Emphasis mine

The Greek aorist passive has no precise equivalent in English, and this present perfect construction was particularly useful for verbs that presented an ongoing state resulting from a past action like ἐγείρω:

A. to arouse from sleep, to awake
B. to arouse from the sleep of death, to recall the dead to life
blueletterbible.org

Some have tried to parse this as a simple present passive construction, but that is problematic. To distinguish it from the predicate adjective with a past participle, the simple present passive normally demands an explicit agent :

Simple Present
Active : Once a week, Tom cleans the house.
Passive: Once a week, the house is cleaned by Tom.
englishpage.com
Emphasis mine

Regardless of the parsing, the expression is designed to emphasize--for theological reasons--the present state of a past action. Orthodox Christians are taught the preeminence of the historical resurrection of Jesus Christ as the foundation of his ultimate sovereignty and their eternal hope, impregnating the entire antiphonal greeting with theological significance:

Christ is risen!
He is risen indeed!
Alleluia!

Unbelievers can find solace in the fact that this greeting is rarely used outside of the church building.

It is correct Early Modern English meaning "He has risen". In older novels one can still find similar sentences, such as "He is come to see you, my Lord."

Present perfect is a phenomenon that emerged in / has spread over many European languages. I believe initially the construction was restricted to certain verbs. In any case, it originally used either to have or to be as the auxiliary, depending on what made more sense for the (full) verb in question. Examples for this:

• He has bought a coat. (He has a coat because he bought one.)
• She has seen the cat. (She has a cat in her mind because she saw one.)
• I am gone home. (I am home because I went there.)
• He is risen. (He is upright - or in heaven - because he rose.)

This selection between to have and to be can still be observed in many (most?) of the European languages that have a present perfect. E.g. in German, Dutch and French (and probably many others), the rule is approximately that to be is used for verbs of motion and to have for all other verbs. The details differ between these languages (and also between northern and southern speakers of German). To have is slowly taking over one after another of the verbs whose present perfect was originally formed with to be. English is one of the few languages in which this process has already been completed. See Wikipedia on Perfect constructions with auxiliaries for details.

(Interesting detail: In some variants of South American Spanish there is a recent development towards using the full verb tener instead of the auxiliary haber, which can no longer be used to express possession.)

It is not contemporary English, as others note in their answers, but mixing present and past tense is not the problem. There is no past tense in the form. I suppose you're taking "risen" to be a past participle, and it is, but despite the name, a "past participle" is not past. It's misleading terminology -- don't let it confuse you.

The past tense (perfect) would be he rose. But this is confirming Jesus' present state of being a risen person. Therefore in this use I wonder might it not be an adjective.