Minimum wage is a price floor. Raising it does not cause inflation. There is no such thing as cost-push inflation. If there were, rising energy prices would have pushed up wholesale prices and prices of goods that use petroleum as a feedstock, and they didn't - intermediate and finished goods makers just saw their margins squeezed and found ways to cut other costs and/or suffered through it, which is why chemical stocks went down and agricultural stocks went down until Bush started talking about ethanol.
Raising it also does NOT give anyone a raise, nor is anyone paid $5.25/hour who would be paid $3/hour but for the present minimum.
Minimum wage works like any other price floor - it renders illegal the sale of a good or service below that price. That means if the market value of your labor at a particular task is $3/hour, and the minimum is $5.25/hour, you don't get hired to do that task - or, you get hired under the table.
At $5.25/hour with all the exceptions they have, the minimum wage is now economically meaningless.
And that is why less than 2% of workers MAKE minimum wage, versus close to 4% when it was set at this level.
$7 is probably meaningless for most people too - McDonald's pays more, Starbucks pays $2 more plus benefits, to START.
Minimum wage is a starting wage, generally earned by students, housewives and recent immigrants - not all but most. This is why 70% of minimum wage earners will receive a 30% raise within a year. There is also incredibly high turnover - most of the one who don't get a raise from the same employer go to another employer, which one is the cause and effect not being quite clear.
And THAT is why "most people don't lose their jobs" when you raise the minimum - - they raise it 75 cents effective six or nine months later or phased in over nine months, and they have a dozen or so exceptions covering most of the jobs you think of when you think of minimum wage. By the time the hike goes into effect almost nobody who was making minimum wage when you debated the hike in Congress is still making it and most of those who are are subject to one of the exceptions.
But it does freeze out of the workforce anyone at the margins.
So the most you can say for it is that in the past, because the hikes have been economically meaningless, not much has happened.
But if you want to know "what would happen" if we raised it by a material degree, consider other countries where it is much higher - - France and Germany typically run 2X our rate of unemployment.
8-9% unemployment is darned good for those countries.
When we have 6% people scream bloody murder.
We have 4.5% now and people aren't happy with it.
But now I say screw it, I'm sick of explaining it to people - let's just set it at $11/hour, watch unemployment go to 8-9% and then maybe they'll believe it.