While it is possible, i don't think it matters Google is all about relevance they will not decide that one site is better than another because it does not have a dedicated IP. But i am very keen to hear other thoughts on this.
A dedicated IP has no effect on your rankings. If it did, 90% of the sites on the web would not rank where they do since MOST small sites, blogs, etc. are hosted on shared IPs at places like GoDaddy, HostGator, etc. Dedicated IPs IMO are, however, totally worth the $2 or so per month extra that most hosts like GoDaddy charge for them for reasons OTHER than rankings. There are occasionally cases where using shared IP hosting can lead to your site accidentally being penalized. A dedicated IP eliminates the risk of this happening because of someone else's actions. For example, sometimes someone on the same shared IP might build a big link farm... If Google notices a significant number of domains on the same IP participating in the the link farm they will sometimes penalize the entire IP. So your domain gets penalized as well... a guilt by association kind of penalty. Of course, once you submit a request for reconsideration and they review your particular domain they will lift your penalty. But having a dedicated IP can prevent you from having to deal with the hassle and downturn in traffic during the time your were penalized. If you have a site that you depend on for making money then $2 per month in extra hosting fees to get a dedicated IP is a very small price to pay for the peace of mind.
Google does look at the IP address 1 site linking to more then 1 site on the same IP is a red flag Google looks at link relationships to find cheating Think about it if you where a Google engineer won't you add this to the algorithm Canoical gets it right with the accidental or nonacidental penalties from competitors go to any highly paid keyword IE loans and you'll see most of the sites on page 1 have dedicated IP's the pros do this for a reason
I realise that less weight is given for links between sites hosted with the same company. But google would be stupid to penalise people for it. I have around 50 websites all related to each other and linked across. It would be unfair to penalise me for specialising in one area and providing content of the same nature.
^ OK if you guys have money, would you go & buy dedicated IP for your site or you just neglect this thing?
Definitely you should go for a dedicated IP if you have the money to do it. It will help in SEO and as the others have said, avoid penalisation to the same shared IP.