@Tyler Speelman as a real estate broker I would advise you to make the release of the high earnest money conditional upon your acceptance of an inspection, conditional approval from underwriting etc.

So what happens if I make the offer, it gets accepted, and then the lender says 'no' to the financing>

The seller accepted.

Does anybody know what may happen if the landlord sends a notice to quit by certified mail that the tenant refuses to accept?

Whatever you decide to do, I'd recommend you do NOT accept payments. 

People selling accept no down owner financed deals for two reasons; either their property has some negative attribute that makes it non financable, or they want, need, or desire to sell at a price greater than the market is willing to pay.

After the 3-day notice to pay or quit has lapsed, you do not have to accept rent.I would recommend working with your attorney through the full process. 

They accepted my offer over others because of strong finances and ability too close, but now are requesting a loan commitment letter even after contract has been signed and earnest money deposited. 

Right before Thanksgiving, we got an accepted offer!

If listing after June, be prepared to either be competitive on price or be willing to wait a long time for an acceptable offer.