我理解此题C是最佳选项，因为英语通常不说found the lamp being on 而可以说found the lamp on. 如果是we found the lamp on but the door shut.是可以的。这可能就是为什么出题人没有用on, shut做选项。

再次查阅了一下朗文和牛津词典，两本词典对find一词的用法解释基本相似，观察所列举的用法条目，似乎可以找到一些思路。

1.《朗文高级英语词典》对find一词的释义及例句：

to discover that someone or something is in a particular condition or doing a particular thing when you see or touch them.

1) find somebody/something+adjective

I’m sure we’ll find her hard at work when we get home.

He tried the door and found it unlocked.

2) find somebody+prepositional phrase

She woke to find a man by her bed.

3) find somebody/something doing something

Often he found her quietly weeping alone.

4) find (that)

She looked at her glass and was amazed to find it was empty.

2.《牛津高阶英语词典》(纸质第9版) find词条7（为简练计，只列举用法条目，不列举例句）:

1) find sb/sth/yourself+adv./prep.

2) find sb/sth/yourself+adj.

3) find sb/sth/yourself doing sth

4) find (that)…

通过对朗文和牛津词典的例句观察，find sb/sth+adj./adverb/prepositional phrase/doing应该可以和find (that)进行同义转换，用that从句体现出一般体意义和进行体意义，或者静态意义和动态意义。以朗文词典的例句为例，应该可以这样：

1) I’m sure we’ll find (that) she is hard at work when we get home.

2) He tried the door and found (that) it was unlocked.

3) She woke to find (that) a man was by her bed.

4) Often he found (that) she was quietly weeping alone.

5) She looked at her glass and was amazed to find it empty.

如果按照这个思路解释:

C项：we found the lamp burning but the door shut.

可以转换为：

We found (that) the lamp was burning but the door was shut.

A项：we found the lamp being on but the door shut.

会转换为：

We found (that) the lamp was being on but the door was shut.(恐怕不能成立）

所以，倾向于选择C项，不选A项。

个人理解，未必正确，仅供参考。

同意你的看法，这道题选 A 也完全可以。

虽然找不到完全相同的权威句子，但类似的例子还是很多的。如：

She found the film being a true story. 她发现这部电影是个真实的故事。

在袁懋梓编著《大学英语语法》（外语教学与研究出版社）第230页找到这样一个句子：

He put on his socks, with the wrong side (being) out. 他把袜子穿反了。

这里的 being out 也是宾补，但这里的 being 是可以省略的。所以这是不是意味着网友的这道题也可以选 A。

两位老师都做了精彩的解答。我这里从考试的角度补充一下这道题为什么答案是C。因为要贴图，所以开贴回答。

既然是考题，就要从考试（textbook）而非实际（real life）的角度出发来考虑，也就是所谓的考点。这里涉及两个方面：

1) 考察find的使用—如果是“主谓宾宾补”的句型（SVOOC），有四个使役动词充当谓语动词的话，宾补有动词的话，要使用该动词的分词形式，也就是在Ving和Ved两种形式种选一个，主动用Ving，被动用Ved。find就是四大之一，也就是说，考试时，你根本不用去看懂整个句子，发现谓语动词是find，而宾补要选一个动词，你直接根据主/被动，在选项中找-ing或ed形式的动词就ok，所以从形式上，D选项就可排除（因为on）；

2）习惯搭配。也就是所谓的collocation。与lamp搭配的常用动词是burn，而且用主动而非被动（be burned）形式，所以宾补选burning (这也是为什么不选A，因为being是多余的）。而门关上了，当然用被动，所以宾补用shut（此处shut不是动词原形，而是shut的过去分词）

以上是基于考试做的分析，脱离考试这个范畴，我的解答可以直接忽略

一家之言，仅供参考；

勤查英英，必有所获！