I guess, to put it succinctly, the law of supply and demand applies just as strongly to employment as it does to anything else, and those who work to subvert it do so at their own peril. In other words, if you raise the price of something (for example, an employee who mops floors at a fast food joint), you get fewer buyers (people who will hire floor moppers).
One objection is this: you're transferring wealth from one person/group to another without receiving any benefit in exchange. Pretty much any time you do that, you lose economic value. Using static analysis (i.e. assuming the business doesn't adapt), the employee gains, but some combination of customers, other employees, or investors lose out. It's a zero-sum situation. At best.
A second major objection is that it removes a couple rungs from the ladder to success. It makes it harder for people with fewer skills and less work experience to get a job. At $15/hour, that floor-mopping job is going to attract not just the teenagers looking for a first job, but also college students, retirees looking for some side income, and other folks who would not have applied if the job paid $7.25/hour. Out of all those applicants, the teenager is probably going to lose out.
Here's a third objection: it creates an even greater incentive for under-the-table jobs. That means lost tax revenue, less safe working conditions, etc.
And one final objection: it removes an incentive to climb the economic ladder. If I can only make $15k/year mopping floors, I'm more likely to invest the time/money to learn a trade that will pay me $35k/year. On the other hand, if I can make $30k/year mopping floors, trade school becomes a lot less enticing.
But does that future come eventually anyway, even without raising the minimum wage? At some point it is simply cheaper to have a couple employees to maintain some computers than to worry about a larger staff. Raising the minimum wage isn't going to prevent that. Hence, something else needs to happen, possibly in addition rising wages.
I think you're on to something there--raising the minimum wage provides a tremendous incentive for companies to automate jobs. Yes, I think that automation is inevitable, but hiking the minimum wage accelerates that transition, rather than allowing it to happen more gradually and naturally. McDonald's isn't the first in the food service industry to do this. I've seen plenty of casual dining places with tablets at each table where customers can review their bill, pick a tip, and swipe their card. Voila, you can now let 30% (or something like that) of your waitstaff go, because they can each wait more tables.