please keep references and comments more concise than the above.  while i appreciate the thoroughness, the formatting of the site makes long comments like that very difficult to read

お炊き上げ：
It is not quite trust-able but it is in the ピクシブ百科事典 definition of お焚き上げ：
http://dic.pixiv.net/a/お焚き上げ
It is also used inside a definition of the 日本辞典：
http://www.nihonjiten.com/data/16731.html

Usage examples:
1) http://www.youtube.com/watch?v=vx31ZkIcCLI
2) Formal homepage of the "Do home Ltd" company: http://event.4-seasons.jp/sys/doc/3023.shtml
3) The homepage of the "Yachi Shrine of God War": http://www.yachi-hachiman.jp/portfolio/osaitou/
4) A blog it explicitly refers to it saying 「今年の正月飾りと一緒にお炊き上げに出した。」: http://sion299.cocolog-nifty.com/blog/2014/01/post-fd40.html
5) Homepage of the "Nichiren cult": http://temple.nichiren.or.jp/6011134-myouhonji/2013/12/id127/
6) Here it refers about how even though the notation is wrong it is still used:
「検索の履歴を見ると。「お炊き上げ」と検索している方も多いですが、正しくは「お焚き上げ（お焚上げ）」です。」
http://moshimo-box.com/ziten.html
7) The Hodosan Shrine use it in the ~official~ name of its お炊き上げ祭：
http://www.hodosan-jinja.or.jp/matsuri/index.htm
8) As it is also indicated in Wikipedia:
http://ja.wikipedia.org/wiki/宝登山神社

お炊上げ：
1) Homepage of a Shrine 「…、これは精進潔斎した神職が忌籠して奉仕し、お炊上げして、新調のお櫃に納めて神前に供するもので、…」: http://hakonejinja.or.jp/02-contents/02-main/08-keigai-jinja/03-contents/03-kuzuryuu/03  kuzuryuu-tukinami 1.html
2) http://okwave.jp/qa/q1744358.html
3) See the comment in the video 「正月飾りをお炊上げして、その火でダンゴを焼いて食べると、１年間無病息災という行事­です。」http://www.youtube.com/watch?v=sMhiEHkPC_8&list=UUjFW9pjHTvMbYsiswZnUEYA&index=15

おたき上げ：
1) 実用日本語表現辞典
http://www.practical-japanese.com/2011/08/blog-post_3530.html

2) Weblioシソーラス
http://thesaurus.weblio.jp/content/おたき上げ

3) An article on SPORTS NIPPON NEWSPAPERS 「陸前高田の松をおたき上げ」
http://www.sponichi.co.jp/society/news/2011/09/26/kiji/K20110926001698600.html

4) TOKYO NEWSPAPER 「被災松　成田山へ　来月「おたき上げ」で使用」
http://www.tokyo-np.co.jp/article/feature/tohokujisin/list/CK2011081502100003.html

5) YOKOSUKA KEIZAI NEWSPAPER
http://yokosuka.keizai.biz/photoflash/945/

The notations お炊き上げ；お炊上げ；御炊き上げ；御炊上げ ~are~ technically wrong and irregular, but in the end still quite used by people, as you realized, almost exclusively in the meaning of　お焚き上げ. The many references I inserted further confirm this evidence. There is also an extra writing I didn't notice before, "おたき上げ", I added it and posted some references to this one, too.

~Very~ seldom the notation 炊き上げ could also be used in its literal meaning, however all over the net I only found one reference to this special case and it would be inferred from the 炊き上げる entry Jim proposed:
1) http://www.amazon.co.jp/ホリカフーズ-ピーエルシーごはん-炊き上げ一番1-25/dp/B001HHIFZW

The last word to you.

@@ -8,0 +9,3 @@
+</k_ele>
+<k_ele>
+<keb>たき上げ</keb>

Koj, Daijr/s (only たきあげ), ＫＯＤ追加語彙 (おたきあげ)
http://www.iromegane.com/japan/culture/otakiage-burning-talisman/

(a) no reference quoted includes 炊き上げ, etc. form, which has a quite different meaning.
(b) we usually do entries in the plain form if the references support this.
(c) proposed wording strayed from that supported by references.

@@ -5 +5 @@
-<keb>お焚き上げ</keb>
+<keb>焚き上げ</keb>
@@ -8,25 +8 @@
-<keb>御焚き上げ</keb>
-</k_ele>
-<k_ele>
-<keb>御炊き上げ</keb>
-<ke_inf>&iK;</ke_inf>
-</k_ele>
-<k_ele>
-<keb>お炊き上げ</keb>
-<ke_inf>&iK;</ke_inf>
-</k_ele>
-<k_ele>
-<keb>お焚上げ</keb>
-<ke_inf>&iK;</ke_inf>
-</k_ele>
-<k_ele>
-<keb>お炊上げ</keb>
-<ke_inf>&iK;</ke_inf>
-</k_ele>
-<k_ele>
-<keb>御焚上げ</keb>
-<ke_inf>&iK;</ke_inf>
-</k_ele>
-<k_ele>
-<keb>御炊上げ</keb>
-<ke_inf>&iK;</ke_inf>
+<keb>焚上げ</keb>
@@ -35 +11 @@
-<reb>おたきあげ</reb>
+<reb>たきあげ</reb>
@@ -40 +16,6 @@
-<gloss>ritual burning of money or other objects as an offering to the gods or to deliver a message to the died ones.</gloss>
+<gloss>bonfire often in temple grounds, usu. of charms, talismans, New Year decorations, etc.</gloss>
+</sense>
+<sense>
+<pos>&n;</pos>
+<xref type="see" seq="1271590">護摩・ごま</xref>
+<gloss>ritual burning of money, cedar sticks, or other objects as an offering</gloss>

大辞泉
http://dictionary.goo.ne.jp/leaf/jn2/135648/m0u/

神社のホームページ
http://www.kurumazakijinja.or.jp/h-otakiage.html

左義長 ・その他の呼ばれ方
http://ja.wikipedia.org/wiki/左義長

A common ritual in Shintou. Every year during the 左義長, お焚き上げ is done when all old talismans or "protections", like the arrows, wood tablets, or pocket "お守り" are burnt in 神社 and their ashes are scattered around the houses are a sign of protection. However お焚き上げ is more than that. It can be done any period of the year to deliver a prayer to the gods or a message to the "afterlife". Sometimes people write letters to their died beloved ones and burn them believing the smoke will go high in the sky and reach them in the "otherworldliness".
It this latter case, it is often done privately without going to a shrine.