▲If I am free, I will do it. 可以改成：If free, I will do it.

其中的 If free 为带连词的形容词作状语。又如：

Start when (you are) ready. 准备好了就开始。

Phone me if (it is) necessary. 必要时给我打电话。

Though (he was) intelllgent, he was very poorly educated. 虽然他很聪明，受的教育却很差。

▲原句也可以改成：Being free, I will do it. (分词短语作状语)，与原句的意思一样。

顺便说一句，Being…这类状语也可以表示原因，如：

Being ill, he didn’t go to school yesterday. 他由于病了，昨天就没有去上学。

Being athletic, Tom found the climb quite easy. 由于身体健壮，汤姆觉得爬山很容易。

Being anxious to please him, I bought him a nice present. 由于急于讨好他，我给他买了一件精美礼品。

Being unable to help in any other way, I gaye her some money. 由于无法给她什么别的帮助，我就给了她一些钱。

所以你的句子 Being free, I will do it. 在没有上下文的情况可能有歧义：

Being free, I will do it. 如果有空，我会去做这事的。（=If I am free, I will do it.）

Being free, I will do it. 因为有空，所以我愿意做这事。（=Because I am free, I will do it.）

▲ Once I am tired, I will take a rest.

本句可以改成：Tired, I will take a rest. 两者意思相同。

当然了为更准确，可以改成：Once tired, I will take a rest.