The short answer is No, it won't have a negative effect on the refinancing. In order to finance the purchase, you have to be qualified to carry both mortgages. If you re-finance to a lower rate, then your ability to carry both mortgages gets better. The caveat in this though is that BOTH houses qualify as your residences. If the old house is deemed a non-residence, then the answer will be yes, the purchase will adversely affect your ability to re-finance the already owned house because it will be considered an investment property at that point. Investment properties carry a higher rate than residence properties.
A full analysis of your situation and probabilities will require disclosure of your private information as if applying for a loan. You don't want to do that in a public forum. I would be happy to take care of your financing. You can call me at 408-639-0211 or email me at firstname.lastname@example.org.