Yes, the landlord most definitely can evict you. Unless and until the foreclosure process ends with a foreclosure sale, the current owner of the property remains the owner, with all the same rights as any property owner. The fact that there is a dispute between the owner and the owner's lender does not mean that you can plan to live there for free. Many foreclosures get resolved, some are wrongful or contested, and you have no way of knowing the underlying details.
The process of eviction for nonpayment does not take long. After expiration of the three day notice, the landlord files an Complaint for Possession with the Clerk, and a Summons is issued and served on you. You have 5 days to respond and to pay. If you don't pay, the landlord can get a default judgment, and then a writ of possession is issued, and the Sheriff serves that on you. From service of the Writ of Possession, you have 24 hours to vacate or they will put you out..