What major translations of the Bible are in the Public Domain? Quoting Biblical text is technically a dicey proposition.  While the original manuscripts are obviously public domain, not every translation is.
Indeed, the NIV even posts the following copyright notice on BibleGateway:

Copyright Information
The NIV text may be quoted in any form (written, visual, electronic or audio), up to and inclusive of five hundred (500) verses without express written permission of the publisher, providing the verses do not amount to a complete book of the Bible nor do the verses quoted account for twenty-five percent (25%) or more of the total text of the work in which they are quoted.
When the NIV is quoted in works that exercise the above fair use clause, notice of copyright must appear on the title or copyright page or opening screen of the work (whichever is appropriate) as follows:
THE HOLY BIBLE, NEW INTERNATIONAL VERSION®, NIV® Copyright © 1973, 1978, 1984, 2011 by Biblica, Inc.™ Used by permission. All rights reserved worldwide.
These Scriptures are copyrighted by the Biblica, Inc.™ and have been made available on the Internet for your personal use only. Any other use including, but not limited to, copying or reposting on the Internet is prohibited. These Scriptures may not be altered or modified in any form and must remain in their original context. These Scriptures may not be sold or otherwise offered for sale.

Given this, what significant translations of the Bible can I actually quote at length?
The issue with copyright translations.

One translation that was produced with the specific intention of avoiding copyright entanglements is the World English Bible.  It is modernization of the American Standard Version (ASV) placed into the public domain.  A paragraph from the site's FAQ is worth quoting:


  The copyright laws of most nations and the international treaties that support them are a mixed blessing. By granting authors and translators a legal monopoly (for a limited, but very long, time) on the right of copying and “first sale” of their works, the law makers have made writing and translating very profitable for some people whose works are in great demand. This has, no doubt, been a factor in the creation of many of the good Modern English translations of the Holy Bible that we now enjoy. The problem with this system, with respect to the Holy Bible, is that it has had the effect of limiting distribution of God’s Word in modern languages. For example, I cannot legally post copies of the entire New International Version of the Holy Bible on my web site in a downloadable, searchable, and readily copyable format without the permission of the International Bible Society and Zondervan (copyright owner and publisher). Zondervan won’t grant such permission unless they get a significant royalty (they quoted me $10,000 + $10/copy distributed) and unless I convince them that my Bible search software is “good enough” for them. Needless to say, the Bible search software that I am writing with the intention of distributing as donorware will not come with the NIV.


Further, organizations such as The Gideons International that distribute Bibles must pay copyright holders a fee if they wish to give away a modern translation (such as the New King James Version) rather than the King James with all of it's Elizabethanisms.  The restrictions can also have an impact on smaller scales.  The NASB licence reads (emphasis in the original):


  The text of the New American Standard Bible® may be quoted and/or reprinted up to and inclusive of one thousand (1,000) verses without express written permission of The Lockman Foundation, providing the verses do not amount to a complete book of the Bible nor do the verses quoted account for more than 50% of the total work in which they are quoted.


These restrictions mean it's (probably) fine to quote a passage or two in an answer on Stack Exchange or in an article or book.  But it's not possible to reprint an entire book of the Bible for the purposes of a "Manuscript Study" (AKA, "Communal Discovery Bible Study Method").  While I doubt the Lockman Foundation will have agents tracking down people who print out Mark for their devotional group to mark up, such a practice violates the intentions of the publisher who fronted the money and commissioned the translation.  As Paul says:


  The elders who rule well are to be considered worthy of double honor, especially those who work hard at preaching and teaching. For the Scripture says, “YOU SHALL NOT MUZZLE THE OX WHILE HE IS THRESHING,” and “The laborer is worthy of his wages.”&mdash;1st Timothy 5:17-18 (NASB)


Finally, many translations restrict derivative works (such as the WEB effort and my idea of making a Southern American Version™).  The US laws concerning derivative works is complicated, but creators who don't want to deal with lawyers would be advised to base their work on translations free of copyright restrictions. 

But God's Word should be free, man!

The Hebrew, Aramaic, and Greek texts of the Bible are public domain (and a cornerstone of our cultural heritage).  But it turns out that even the critical text is protected by copyright laws.  For the general public, this is no great loss as there are plenty of unrestricted options and the licences for most translations allow for the most common uses of the texts.  But all of this has serious implications for scholars who honor the wishes of the Bible publishers.  It's tempting to get angry with the publishers or to ignore the restrictions.

However, we have a duty to comply with the licences of our translations so that the hard-working folks who produce them can afford to keep laboring.  Most of them are reasonable people who are as interested in seeing God's Word preached as you are.  In fact I'll let the Crossway (publishers of the ESV) blog have the final word:


  We're not going to come after you if you don't cite quotes from the ESV according to these guidelines. But we'd appreciate it if you did. Using the letters “ESV” also helps us track the popularity of the ESV in the blogosphere.