Even XeTeX isn’t complete, as you can see by scrolling almost to the bottom of this page. It is missing Swash, Ornaments, Slashed Zero, Justification Alternates, Alternate Annotation, Mathematical Greek (??), Localized Forms, Glyph (De)Composition, Required Ligatures, Mark Positioning Via Subs, and Isolated Form.
Thanks for pointing this out, I had forgotten about such gory details :-). This article, and the “ilovetypography.com” site in general, is a great resource.
At least it looks like Caolan McNamara has implemented localized forms separately, as shown by oweng in the comments of my answer.
Here’s a complete Web implementation of OpenType: http://www.impallari.com/testing/ And its source code: GitHub - impallari/Font-Testing-Page: A webpage for testing typefaces, live at www.impallari.com/testing
This feature is extremely important especially for arabic fonts, as they are based on ligatures, glyphs etc.
The “bug ID” is said to be invalid when following the link.
It works now. 
Thanks. This bug has some interesting comments. I agree with one of the commenters: XeTeX has the best OpenType support right now and LibreOffice could definitely build on that. XeTeX’s smart font handling is based a modified version of ICU, which is already used from LibO 4.0 onwards for regexp.
Update: XeTeX is now based on Pango/Harfbuzz, which is the sexier, actively-developed and emerging solution for OpenType support. Possibility: let OpenOffice remain the ICU-using IBM-style suite, and let LibO embrace the future with Pango and Harfbuzz.
@CyanCG, a comment by Caolán McNamara here states, “There is an experimental use-harfbuzz patch that we should get around to applying …”, so I guess you can expect LO to eventually replace ICU with Harfbuzz at some point.
@oweng I’ve submitted a comment at Caolan’s post to which you linked; it’s still awaiting moderation. I informed him of this question and the related bug 58941 to which you linked, and also of the importance of supporting more OpenType features.
@oweng McNamara has posted my comment and responded with “LibreOffice will support them when someone writes the relatively small pieces of code, UI glue and flags to enagle [sic] them.” So, it would appear that this is an easy job.
Does anyone know if there are any updates on these simple patches and harfbuzz support?
Of course it would (will!) be wonderful to have Open Type supported by LO.
In the meantime, Graphite fonts are not to be sneered at (not that anyone here is sneering!) In addition to the complex Roman+ fonts listed at the Graphite site, there are also the beautiful (IMO) and versatile Linux Libertine and Biolinum Graphite-enabled fonts (+ link to PDF documentation).
So, if “Anonymous” original poster is still watching: while waiting for Open Type to become fully supported, at least make good use of such Graphite fonts as are available!
+1. I cannot mark this up enough. In basic terms supporting OpenType = supporting Adobe, which would seem at odds with an open source project like LO. As @dajare mentions the SIL Graphite fonts are exceptionally good. IMO it is better to have one great font than a dozen poor ones.
As I recall the OpenType specification is an open standard, so there’s nothing wrong with implementing it.
It is not that simple. OOXML is “open” too, but using it merely promotes MS and is highly contentious as indicated on the mailing list. It is in the interests of LO to promote platform agnostic solutions i.e., Graphite (over OT / AAT). PostScript outline fonts are not likely to be embeddable within an editable document e.g., ODF. Ask Adobe why that might be.
I don’t believe that implementing OpenType implies being awkwardly encumbered the way implementing OOXML does. The modern TeX engines (XeTeX and LuaTeX) implement OpenType with much success and they are both free software projects (XeTeX was actually started by a SIL employee). Yes, Graphite is an excellent technology that deserves more widespread use: right now, it is not implemented on OS X and I wish it was.
I’m interested: does the Graphite in LibreOffice not work on OS X?