No, it won’t.
For years there’ve been rumors about the negative effect of hosting several websites on the same IP. Although, in the past, an under-populated IP may be held some “extra” SEO value, today this is not the case, partly because of IPv4 IP address shortage which makes the whole idea simple unfeasible .
Google officially addressed this rumor on a several different occasions. Here are some references:
“I wouldn’t worry about being on the same IP address and I definitely wouldn’t worry about being on the same server that is something what everyone does…”
Matt Cutts – Head of Google WebSpam Team
“We generally do not treat sites hosted on CDNs any differently”
Google Official Representative