If Snape was removing his memories of Lily and placing them in the pensieve every time he and Harry practiced Occlumency to keep Harry from seeing them, does that mean that Snape, in essence, could no longer remember them? Does that also mean that if you removed a memory to the pensieve to keep someone from seeing it, would this be a way to avoid the truth telling effects of Veritiserum?
Based on a convincing argument I've decided to amend my answer to say that it must depend on the situation.
Dumbledore showed Harry several of his own memories in the series (meeting Voldemort, trials of Death Eaters after the first war) at times he even accompanied him into them.
Memories like these were stored in his office and not his head, so they must be copies. If he couldn't remember them himself when they were in the bottles/phials in his office or in the Pensieve itself, the situation would have been very different.
Since Snape seems to use the Pensieve for protecting his memories, perhaps there is a way of either copying or storing the original depending on your need. Though, thinking about it, if Snape forgot what he put in there, how would he know to go back for them? Perhaps they are protected in a different fashion in the Pensieve.
Pottermore has JKR's writing on the Pensieve which explains:
There is no mention of hiding memories in the way that Snape does. It clearly is primarily an aid to thinking and studying memories, not something designed to make you forget.
It seems you don't remember memories in the Pensieve. As you say, Snape used this as a protection against Harry during Occlumency practices in Harry Potter and the Order of the Phoenix:
If he did remember the memories, there would be no need to restore them after each practice:
And he wouldn't be that worried about them:
And in Harry Potter and the Goblet of Fire, Dumbledore says he uses the Pensieve when he has too many thoughts:
If the Pensieve didn't remove the memories from your head, it wouldn't help in this situation.