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?
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:
The light was coming from the Pensieve sitting on Snape’s desk. The silver-white contents were ebbing and swirling within. Snape’s thoughts … things he did not want Harry to see if he broke through Snape’s defenses accidentally. … Harry gazed at the Pensieve, curiosity welling inside him. … What was it that Snape was so keen to hide from Harry?
— Chapter 28, Snape’s Worst Memory
If he did remember the memories, there would be no need to restore them after each practice:
As he opened it he glanced back at Snape, who had his back to Harry and was scooping his own thoughts out of the Pensieve with the tip of his wand and replacing them carefully inside his own head.
— Chapter 24, Occlumency
And he wouldn't be that worried about them:
Panting slightly, Snape straightened the Pensieve in which he had again stored some of his thoughts before starting the lesson, almost as though checking that they were still there.
— Chapter 26, Seen and Unforeseen
And in Harry Potter and the Goblet of Fire, Dumbledore says he uses the Pensieve when he has too many thoughts:
“This? It is called a Pensieve,” said Dumbledore. “I sometimes find, and I am sure you know the feeling, that I simply have too many thoughts and memories crammed into my mind.” “Er,” said Harry, who couldn’t truthfully say that he had ever felt anything of the sort. “At these times,” said Dumbledore, indicating the stone basin, “I use the Pensieve. One simply siphons the excess thoughts from one’s mind, pours them into the basin, and examines them at one’s leisure.”
— Chapter 30, The Pensieve
If the Pensieve didn't remove the memories from your head, it wouldn't help in this situation.
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:
"The Pensieve is enchanted to recreate memories so that they become re-liveable, taking every detail stored in the subconscious and recreating it faithfully, so that either the owner, or (and herein lies the danger) a second party, is able to enter the memories and move around within them."
"Even more difficult than the recreation of memories is the use of a Pensieve to examine and sort thoughts and ideas, and very few wizards have the ability to do so."
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.
Perhaps you lose the original memory, but when you view it through the pensieve, you have the memory of viewing it through a pensieve. And the point is that that memory is actually more detailed than your orginal memory.
Further, while memories tend to distort over time, once you put a memory in the pensieve, it won't change.
Dumbledore obviously uses the Pensieve to clear out space in his mind. As he states in the Goblet of Fire. I BELIEVE that the detailed memory is removed, but a memory OF the memory, and the "title", so to speak, of the memory remains. I feel this is the only logic that can account for all of the circumstances in which someone removes a memory. Think of it like zipping a file so it doesn't take up so much room and can fit in an email. The mind retains the zip file while the memory in the Pensieve, or a glass jar, is the expanded version.