Is it OK to leave what's left of a tooth that's now level with the gum in place?

It is OK as long as the remaining tooth is healthy and/or has been restored to health with a filling or crown. Leaving the root in place can provide support to the partial denture and it maintains the jaw bone in this area which is beneficial. If there is decay in the tooth or gum disease around the tooth then it should be extracted.

It is okay to leave it as a support for the partial denture AS LONG as the tooth is not decayed.  If there is any decay in it and the dentist has not filled it or put a covering cap over it my recommendation is to have it removed before it abscesses or else decays below the gum line which would make the extraction much more difficult.What your dentist is doing is a fairly common technique called an "overdenture" (i.e. a denture over a tooth/teeth or implant(s). 

While it may be counter-intuitive, leaving the root in place is a method that has been around for a long time.  If you are keeping the root, it is best to cover what remains with a coping to prevent decay and give reinforcement, but even better is to remove the root and place an implant.  Without the root there is no risk of decay.