Normally you would just look it up for your location--but I'm thinking of a peak where the horizon will be at least a mile lower and I would think that would have some effect.
With some trig and simple approximations, for the case where the sun is rising directly to the east, and is rising in the direction perpendicular to the horizon, the result is
T = (24 hours)(1/2π)√(2h/R),
where h is your height above the spherical surface of the earth, R is the earth's radius, and T is the amount of time, in units of hours, by which sunrise comes early. This might be a reasonable model for cases like the eastern Sierra or Kilimanjaro, where you have a mountain range that is fairly prominent relative to low-lying terrain to the east. It might not make sense, e.g., for ranges that run east-west, such as the San Bernardinos or the Himalayas. Then there would be more chance of having other heights to your east that are not much lower than you, or maybe even higher.
Putting in numbers for Kilimanjaro, I get 10 minutes.
For a one-mile tower, the result is 5 minutes.
It is not normally true that the sun rises straight up and to the east, so the result will really only give a rough idea of how big the effect is. At extreme latitudes, the effect could be much bigger. If you're above the arctic circle, the sun could come up in the spring a week earlier because you're on the top of a mountain.
The experience I have had much more often is that the day before summit day, I'm camping in some valley near the peak, with tall mountains on all sides. Then it gets dark fairly early as the shadows sweep across my position, but I can still see the bright sky and the sun lighting up the peaks to the west.
The question asks for first light but let's assume the time difference will be the same as for dawn. And also that the terrain is flat, apart from an observer in a tower 1 mile high, and the radius of the Earth is 3959 miles.
Here is an exaggerated geometry sketch showing a horizontal line in the direction of the sun for an observer at ground level, and a diagonal line showing where the sun will rise for an observer on the tower, which is tangential to the Earth's surface.
The difference is the angle A which is the same angle as subtended by two radii from the tangent points.
So we have cosine(A) = (3959 / 3960), giving A = 1.28 degrees.
It takes the sun 24 hours to move through an angle of 360 degrees, from this can be calculated that the sun will rise 5.12 minutes earlier (or set 5.12 minutes later) than for an observer on the ground.
So about 5 minutes.