It depends on the sentence. Consider this one (submitted by a student of mine):

"Thomas had been perfectly willing to look the other way while men such as I lived our lives the way we wanted."

In this example, I lean towards the objective case. Does “such as” act as a linking verb? I don’t think so. It’s a preposition, no? (Sometimes.) Shouldn’t what follows be the object of the preposition? Another way to think about it: If you put “like” in place of “such as,” would you say “I” or “me”? I’d say “me,” in the same way that you'd say "while men under me raised their eyes." So I advised the student to replace "I" with "me." (He probably won't. He's very stylistic.)

For other discussion on the question, see the somewhat grumpy answers by Colin elsewhere on this site.

It depends on the sentence. Consider this one (submitted by a student of mine):

Thomas had been perfectly willing to look the other way while men such as I lived our lives the way we wanted.

In this example, I lean towards the objective case. Does “such as” act as a linking verb? I don’t think so. It’s a preposition, no? (Sometimes.) Shouldn’t what follows be the object of the preposition? Another way to think about it: If you put “like” in place of “such as,” would you say “I” or “me”? I’d say “me,” in the same way that you'd say "while men under me raised their eyes." So I advised the student to replace "I" with "me." (He probably won't. He's very stylistic.)

For other discussion on the question, see the somewhat grumpy answers by Colin elsewhere on this site.