The executor has the right to manage the property of the estate. However it seems odd that he would want to evict someone, only to transfer the property to them later. The will provisions may even provide a defense to the eviction, since the person who is to inherit the property would have equitable title.
I can't think of any reason he would want to evict the person who is to interit the house, especially if they're paying the bills and caring for the property in the meantime, so I suspect there may be more to this story.
The executor was appointed by the court to manage the estate's assets including the house. It does seem weird that he would want to evict the person inheriting the house in the interim. If you question the eviction decision, then file a motion with the court to decide the issues. You will need a probate attorney to help you.