Kind of a rookie question here but do any of you require a new background check when renewing your tenant’s lease?

We don't do another background check but we do an inspection of the property to see if we want to continue renting to that tenant. 

Nope, if they are paying on time and taking care of the property then that would not be necessary. I would do at least an annual inspection before renewing. I have quarterly inspections written into my lease, but I dont perform them because I do repairs myself and I am in the units every so often and check the condition at that time.