I am assuming that ship has sailed by now since you are close to closing but something to think about for the future.Contact a lawyer and see what the best way to go about evicting a month to month tenant is in your state.For instance, in CT it is much easier to evict for non payment of rent as opposed to lapse in time.  

The new tenant moves in with no lapse in rent, we don't have to pay $100 for initial treatment because it's under control.  

I would say it's required, and you have to be listed as someone who gets notified if their insurance lapses, and if it does, you can give them a notice to cure or quit.  

You can and may require that tenants list you as an "additional interest," which would give you notification of cancelations, lapses, non-renewals, and renewals.Additional interest doesn't provide you any coverage, of course, beyond the liability coverage on the policy, but it certainly puts your mind at rest.  

They are going through a divorce and they have let things lapse.  

I dont always act on the notice right after it lapses but if i need to, the notice has been served as early as possible.

That way you wont have any lapsed time for that valuable knowledge you will likely never use to fall out of your skull.  :-)Good choice with Micki, shes great!

Then they decided to not pay 2 of my houses insurance and let them lapse.  

There isn't a lapse in pay, but there is a delay while the paperwork is completed. 

Most of these Landlords probably allowed the lease to lapse into month-to-month because they're too lazy to renew.