Does my site need a dedicated IP address?

Perhaps. If you have a dedicated IP address, you will get some advantages. Thus, your site can be accessed by IP-address without waiting for domain name registration (unless, of course, the provider will give you a temporary domain level 3). It will be possible to install SSL. And then, if your neighbor on the physical server will be a resource that is banned by search engines or other services, then your site will also be unavailable if you do not have a dedicated IP.