Since I am copying file from production server, I do not want to
affect the performance if scp takes time. Does this operation affect
the performance of remote server?
Define "performance." If we're talking CPU performance, then SCP is going to cost you the price of encrypting with 128-bit AES (I believe that's the default for SSH2, but can't find a reference right now).
If you're talking bandwidth performance, then it will cost you the overhead of your encryption algorithm and strength.
If you're talking throughput performance, then you'll have to rate limit to your needs using either
limit in SCP (as David Schwartz mentions in his comment above) or an different tool like perhaps
Is there any better / faster way to copy a file? The file is already
going to be different (name based on current date and time)
Define "better" and "faster". Define them very carefully. If you're saturating your CPU (very unlikely) there are different options than if you are saturating your bandwidth. Both involve rate limiting, however, and are fairly straightforward.