Will going 2 miles in 60 degree weather cause damage to new winter tires?

Your friend is an idiot. What does your friend expect will happen to your tires, if you put winter tires on your car and you live in a part of the county where it's 60 degrees for over a week, then it drops down to around 30 degrees for a few days,then warms back up again to 70 degrees, and then a couple of weeks later gets so cold that you have snow and ice on the road? If "winter tires" are so fragile that would harm them, then don't put them on your car at all.

You can't be switching your winter tires for regular tires every other week.

Your "friend" is an IDIOT!

Damage? The tires are made for Winter driving, not cool weather. Anytime you just put tires on a car, it begins to wear. Winter tires give you better grip on the road, but in warm weather that's not needed. You're wasting the tread life.

Your friend is an idiot.

Jeez, where do these people come from? Yes, they might melt.

Your friend is an idiot who knows nothing about winter tires.

It will not.       

You will get faster wear on warm, dry pavement - the tread rubber of winter tires is considerably more flexible than that of all season and summer tires. That same pliable tread rubber that adds traction in winter will wear down quickly in warm temperatures. ... 2 miles isn't going to hurt them.