Would mongodump cause replica set to crash?

Running backups on a Mongo Replica set - 1 primary, 1 secondary, 1 arbitrar. Had a instance where all three stopped talking to one another and all went to secondary.

I ran a backup against the the Replica set with Mongodump. Would that cause the DB's to switch? Is there harm in running mongodump against a running production system on the Primary?

No, there shouldn't be any harm in it. Producing the dump will take some server resources, naturally. Can you check the logs of the replica set members and see what they say around the time the set went read-only, and post it here? That will tell tell us why the primary stepped down. Also, I'm assuming the arbiter didn't turn into a secondary - you just mean that there was no primary? It would also be helpful to see what was logged by mongodump. Please also include the version of MongoDB you're using.