Question: We have an outdated build of SQL Server 2008 R2(SP1) and therefore we are in need of patching(SP3).

Does this have any impact on SharePoint? 

What measures should I take, besides backing everything up and testing before doing in production?

Answer: 
SharePoint 2010 is supported on SQL 20018 R2 SP3, so no, there shouldn't be any issue installing the service pack onto your SQL Server.

I'd test (if at all possible) on a test system first, simply to ensure that any other solution installed within the SharePoint farm also continues to function as expected once the service pack is installed.

Obviously you should also take a set of backups of all of the databases on the SQL Server before you proceed (and test the backups!) in case you have any issues and need to be able to roll back. A SQL alias (as mentioned by Benny) is your friend here as you can run up a new server quickly, change the SQL alias(es) to point to the new server and you're good to go.

During the installation of the service pack there will be downtime to the farm as the database services will be unavailable for a period of time during the patching process. If at all possible, shut down the SharePoint server before starting the work.
Upgrading / Patching of SQL server is not big issue as compare to patching the SharePoint server. You have to plan it properly.

* First thing, you have to apply the patches in lower environment and test the behavior of both SQL as well as SharePoint. Once satisfied then move to production.

**For Production**

* You have to schedule a downtime as it will interrupt the sharepoint. off hours always great.

-Backup the SQL Server( all the databases)
* Shut Down the SharePoint servers, I do this way but if you dont like it then stop the timer & admin services, IIS, Search service.
* Apply the patches to SQL server.
* Verify the patches installation.
* Bring the SharePoint server back.
* Test the Farm, also check the event log & ULS for unusual activity.
* Run the Full backup of the SQL again.