Of your three options, the first is wrong. You cannot say: He was seen do the test. If you use the verb do in this context, it requires the infinitive to in front of it.
Thus, it's correct to say: He was seen to do the test. This implies - without stating explicitly - that he was seen to have taken the whole test and completed it.
It is also correct to say: He was seen doing the test - using the participle rather than the infinitive. This states that he was seen in the act of doing the test, as though somebody had glanced through a classroom window and seen him at the desk.
Unlike the previous example, it does not imply that he was seen completing the test.
However, neither construction is explicit. A listener would not know without further information whether he had completed the test or merely been witnessed in the act of being tested.