Is it appropriate to capitalize pronouns referring to Jesus Christ?

I have been taught (in ELCA seminary) that the word LORD, presented in the Hebrew Bible in small caps, represents the ineffable four-letter holy name and so should be capitalized. I was also taught that God is a proper name, as are Jesus, Christ, and Holy Spirit.

However, I've also been taught that pronouns referring to the unity or diversity of persons of the Trinity are not to be capitalized. @Peter Turner noticed the same in Roman Catholic writings. It's the same for my church body.

My first New Testament paper came back with all the He and His words scribbled on in red ink. Being a good little grade-grubber, I changed my ways.

I understand that it is done out of some idea of respect, but personally, I think that for pronouns it is just bad grammar. Capitalizing titles and proper names for God and Christ is, however, correct application of the rules of English grammar.

Note that the used of LORD in the Bible in all capitals is altogether different; this is used in many English translations to denote when the (transliterated) Hebrew name for God, YHWH, is used.

Letter case is a relatively modern innovation, so we can't go by the original languages. Biblical Hebrew, Aramaic and Greek were all written in what we would call uppercase. They also lacked other typographic conventions such the modern system of punctuation. (You can get an idea about what ancient writing looked like by reading this history of the pilcrow.) The primary purpose of case is to aid the reader, so different languages have different conventions. In German, every Noun is capitalized. In Spanish, days of the week and months are not capitalized. In English, personal nouns are capitalized and pronouns are not. This is a matter of language convention.

Where things get interesting when we enter the territory of capitonyms: words that may be capitalized or not depending on the intended meaning. One example of such a word is god, which is usually capitalized when referring to a (or rather, the) monotheistic God. So one might refer to either "the Greek god of war" or "the Hebrew God". Frankly, the capitalization need not signify respect or honor in this case; rather it is a sign of uniqueness. "God" functions like a personal name in this instance.

I have not been able to find out how pronouns referring to God began to be capitalized, but it does aid in the understanding of some Biblical texts:

When the LORD saw that he turned aside to see, God called to him out of the bush, “Moses, Moses!” And he said, “Here I am.” Then he said, “Do not come near; take your sandals off your feet, for the place on which you are standing is holy ground.” And he said, “I am the God of your father, the God of Abraham, the God of Isaac, and the God of Jacob.” And Moses hid his face, for he was afraid to look at God.—Exodus 3:4-6 (ESV)

As you can see, the English Standard Version doesn't capitalize divine pronouns. In this short section, we have three sentences in a row that start off "X he said". Two of the 'he's are God, which is clear enough from what is being said. (Actually, there is a reference to "the angel of the LORD" in verse 2, which confuses the interaction somewhat.) We can see that the exchange is a bit more clear if we use a translation that capitalizes "He":

Exodus 3:4-6 (NASB)
⁴ When the LORD saw that he turned aside to look, God called to him from the midst of the bush and said, “Moses, Moses!” And he said, “Here I am.” ⁵ Then He said, “Do not come near here; remove your sandals from your feet, for the place on which you are standing is holy ground.” ⁶ He said also, “I am the God of your father, the God of Abraham, the God of Isaac, and the God of Jacob.” Then Moses hid his face, for he was afraid to look at God.

Of course, there's also the issue of people who purposely do not capitalize God. This is pretty common among atheists. Obviously, they would not bother capitalizing "he" at all. Why would they? When interacting with atheists, I try to avoid capitalization unless it helps make my point clear. That can sometimes mean leaving God lowercase when it might confuse matters. I don't think He minds.

When it comes to Jesus, it probably doesn't pay to overthink it. (But watch me!) One rule might be to switch the case depending on whether his humanity or His deity is in view. But unless that's something you are writing about, you're more likely to confuse your readers than edify them. Better is to pick a convention and stick to it. Don't forget Jesus' warning:

And he said to them, “Well did Isaiah prophesy of you hypocrites, as it is written,

  “‘This people honors me with their lips,  
          but their heart is far from me;    
    in vain do they worship me,  
          teaching as doctrines the commandments of men.’  

You leave the commandment of God and hold to the tradition of men.”—Mark 7:6-8 (ESV)

Summary

Capitalization of pronouns is a convention of language and not intrinsically a sign of honor or respect or divinity. There are good reasons to use "He" and equally good reasons to use "he" to refer to Christ depending on context.