You could word that question as:
Does an increase in the minimum wage cause an increase in the cost of goods and services?
And the answer to that depends on who you ask.
Employment Policies Institute (a conservative think take financed by the restaurant industry lobby) says yes: https://www.epionline.org/studies/r100/
Using a variety of government and private datasets, Aaronson and French show that prices do in fact rise in response to a minimum wage increase. Aaronson (2001) finds that minimum wage increases tend to raise prices. The magnitude and timing of these price increases is striking. Within three months of a wage hike, Aaronson finds that a 10 percent increase in the minimum wage resulted in a 0.4–0.7 percent increase in restaurant prices. Much of the increase occurred within the first month of the wage hike. In the fast food sector, prices rise 1.5 percent in response to a 10 percent increase
Truth-Out (a progressive/liberal news publication) says no (albeit it's caveated by saying it's very tiny: 1/10th of 1%): http://www.truth-out.org/news/item/14050-minimum-wage-hikes-do-not-cause-inflation
This fear of inflation from the minimum wage is not based on any reasonable description of how these minimum wage hikes will likely impact businesses, or the economy more generally. The potential impact of minimum wage hikes on the overall price level is simply too small to have any appreciable impact on inflation
Arthur Macewan (professor emeritus of economics at UMass-Boston) offers up a more nuanced not very much answer: http://dollarsandsense.org/archives/2014/0714macewan.html
So what happens when the minimum wage is increased? Consider the situation in restaurants that have a middle-income clientele. Restaurant owners, facing a higher wage bill, would like to pass the costs on to their customers. But their customers are people whose incomes have not been affected much, if at all, by the higher minimum wage. So if prices at the restaurants go up, these people will buy less and the restaurants will now lose some profits. They may raise prices a bit, but not much. Whatever they do, the restaurant owners will have to, if you’ll pardon the term, eat some of the increased costs.
And, in the end, I think Vice (a--well--Vice is Vice) sum it up succinctly: https://news.vice.com/article/what-will-happen-to-the-economy-if-we-raise-the-minimum-wage
The economics of the minimum wage are complex enough, and the historical record inconclusive enough, that economists can make arguments either way. So they do.
I tend to agree with Vice. Economics is complex and the entire minimum wage debate is huge. There's all sorts of factors involved and one has to really look at the debate in context.
PS: Just one example (not saying this is the norm--just that it's interesting and timely), a restaurant in Seattle has recently figured out a way to raise minimum wages to $15 but not raise prices overall by increasing the cost of meals, but doing away with tipping: http://www.king5.com/story/news/local/seattle/2015/07/31/ivars-salmon-house-minimum-wage-tip-service-charge/30968531/ (so, those that used to tip generously are now actually paying less, while the cheap people out there, alas, need to pay a bit more now. :)