Of course it will - if you remove redundant entries and indices from any database it'll speed up access times and that's all the registry is. The amount of speed increase will depend on the number of entries removed. I've seen significant performance changes after running CCleaner.
CCleaner's main function is not a registry cleaner. The performance gains seen after running it are mainly due to it's other functions. The performance increase simply from running the registry is minimal, there would need to be thousands and thousands of broken entries for it to make a difference.
On the other hand, regularly cleaning does result in a fragmented registry, slowing down access.
In defense of CCleaner, it's a good program, and it's registry scanner is more accurate than most.
Most registry cleaners cause more damage than they solve, and quite a few more are just simply malicious. If you are recommending cleaning the registry to someone, they are just as likely to download other registry cleaning tools that are likely to cause damage.
You are also advising a user to touch the registry, who's initial question was "Can I delete the registry". You shouldn't go near the registry unless you understand how it works, and are an experienced user. The many cleaners that advertise "Fix windows errors now!" etc, are quite frankly scams and will just damage the computers of those who blindly download them. Most people don't know what the registry is, but many of them are willing to run cleaners without a moments hesitation.
Edit:
Just to illustrate my point, I just scanned for issues in CCleaner, and found out of the the scan, about 20 "Issues" that were correct registry entries that should not have been removed. Two of them were the uninstaller entries for two working and installed programs. Continuing with the registry clean, this would have broken several file extensions and removed two installed programs from the "Programs" list in control panel.