Dawn, the answer to your question is that yes, you would have to start the process over, but I don't know why you would.
I hate to come on strong on you, but you are in a declining market. If the appraisal came in at $116,100, then you shouldn't be paying more than that-unless the appraiser is way off, which is often not the case.
If you are asking about a HomePath mortgage, that would mean that the house is owned by Fannie Mae (HomePath is only offered on Fannie Mae homes).
They make their guidelines a little easier so that they can unload their homes. That doesn't mean that the house is worth buying.
What would happen if you needed to move and sell the home? Any potential buyer would have to do an appraisal and wouldn't be willing to buy the home from you.
It is never wise to pay more than the value of the home unless you must live in that specific area and can't find a similar home for that price.
You said that you were getting and FHA loan. That means that the seller would have signed an "ammendatory clause" which gives you the right to walk away from the deal and get your deposit back.
If you have to pay a few thousand more then I can understand. If not, you should strongly consider walking away and looking for a different home.
I hope this was helpful.