This really depends on your infrastructure, but the likely answer is No.
- If both the IPs are in the same data center, and that data-center goes down, (or your switch), then multiple IPs will not help you.
- If both IPs are pointing to the same physical server, then that too will not help you. (I doubt this because you said you have to dedicated servers)
You really only need one static IP per server. Possible reasons for having more than one IP are if you are doing multiple SSL sites (to support older browsers each SSL cert must be tied to a specific IP), or if you are running mail servers and want to use multiple IPs to maintain a good reputation if you spam (in which case you should evaluate your marketing efforts and maybe use a service like Mailgun or Sendgrid)
As for fail over you have a few options. You can set fail over at the DNS level. Services like DNSmadeEasy have features where they monitor your site or application, and if it goes down reroute your A records to a different server.
Or you can use a ReverseProxy Server. I don't know what you are running on your server, but HAPRoxy would be a good choice to provide you both fail over and load balancing if it is an email or web application. Some items like SSH or RDP don't work well with HAProxy, but HTTP, MySQL 3306, and SMTP all work wonderfully. HAProxy barely uses any ram, and it too can be setup in a dual node fail over scenario where you really don't have to worry about it going down.
The way we do it for our web apps is we put HaProxy in front of a bunch of Linux and Windows servers. HaProxy routes traffic based on the rules we define. For some of the servers we have fail over or load balancing, where if one back end server goes down then HA Proxy takes that out of the loop until its back up. We also have DNS based fail over in case our single HA-Proxy goes down to route traffic to an amazon server which has a script that notifies us (using Twilio) and lets the visitor know the service is down. We really never had the Ubuntu server that runs HA-Proxy go down on its own. It only really fails over if our hosting provider has an outage or if we restart the server, which we rarely need to do. The HAProxy server is setup with a bunch of IP Addresses, all static, for different web sites or applications. It makes it very easy to manage.
You can learn more about HAProxy on stackoverflow.com