Cluster Validation on Remote Node
I would like to run a validation test on that node (dr site). Will this affect the other 2 nodes which are in synchronous mode?


No not the data flow as such. You can run cluster validation of all the resources but storage with no downtime. Although few of MS blogs suggest you can run all cluster validation test but storage on WSFC without downtime, still I would suggest you to run the validation when load is relatively very less. 

Please note that when you run validation of shared disks cluster manually failover the disks to each node to check the failover and availability and hence it will cause downtime apart from this all other failover tests are online. I assume your AG is one having local disks attached to each node so I do not think you have to worry about downtime but again run validation when load is relatively very less.

I would suggest you to read Validating a Cluster with Zero Downtime. Quoting from the blog


  Across these dozens of tests, only a few of them will impact running cluster workloads and these are all within the storage category, so skipping this entire category is an easy way to avoid disruptive tests.  Listing All Disks and Potential Cluster Disks will not impact anything.  Validating Disk Access Latency, File System, Microsoft MPIO-based disks, and SCSI device Vital Product Data (VPD) can impact the disk’s performance as tests are performed against the disk which is in use, however they will cause no downtime unless the disk latency becomes so slow that it triggers an alert.  Several tests will actually trigger failovers and move the disks and groups to different cluster nodes which will cause downtime, and these include Validating Disk Arbitration, Disk Failover, Multiple Arbitration, SCSI-3 Persistent Reservation, and Simultaneous Failover.  So if you want to test a majority of the functionality of your cluster without impacting availability, exclude these tests.