There are few places currently with a $15 minimum wage so that is a very unlikely cause for the higher prices.
What I have noticed is that prices vary dramatically across the country depending on where you are even among the same chain.
And this: I have been from Maryland to Arizona and then back and forth from Arizona to Indiana 3 times in the last 3 months. I have been to A LOT of fast food restaurants. And one thing is apparent everywhere: They are all hurting from a lack of enough employees. They ALL were advertising for workers. It didnt matter what city or state we have been in they all needed more workers.
In discussion with some of the managers I asked about the effect of requiring drug and background checks. Which has created an entire underemployed/unemployed cohort in our society. In almost every conversation I was told they have dropped those requirements.
The one thing you will see in Fast food places that IS related to the $15 minimum wage is the automated Order Kiosks at McDonalds. They are trying to get ahead of the game by eliminating or reducing the need for order takers. Do everyone a favor DO NOT USE THEM.
I think the rise in prices has more to do with want for bigger margins, low unemployment, higher wages (across the board not just in fast food resulting in more disposable income)and it is clearly what the market will bear.