Only if the attic ventilation is improved when the old shingles are removed and/or more insulation is added in the attic.

In a hot climate lighter color roofing will reduce cooling costs and in a cold climate dark color roofing will save heating costs. This is usually only important in extreme climates.

Actually, in most areas of the Continental US, shingle color has more impact on melting snow than on attic heat if there is sufficient attic insulation and air flow.

If attic insulation and air flow are insufficient, shingle color will make a small difference, but not enough to save much money on heating/cooling.

I have a house with black shingles and the roof is L shaped. Part of the L is a north facing section. The snow on that section is the last to melt. The other half of that section is the first to melt since it has direct all day sun.

The color of roofing shingles could have no effect on heat transfer up from the attic or down from the sun so it should have no effect on snow melting.

If the attic insulation is in the attic floor, the color of asphalt shingles will have more effect on the temperature of the roofing (and therefore the attic floor through radiant heat transfer) than attic ventilation will. See Carl Cash's field study of this issue in his book "Roofing Failures".

Here is a link that might be useful: field study

Heat transfer from the attic has little effect on snow melt as far as shingle color is concerned.

Black shingles get hotter than white shingles. Gray shingles get hotter than white, but not as hot as black.

White shingles with some black granules will cause snow to melt a bit faster than solid white shingles with no heat in an attic.

Is the difference worth the effort? I doubt it. Having sufficient air flow in an attic is much more important that the heat transfer characteristics of shingle color.

For shingles to be warmed by the sun they would need to be exposed; sunlight does not pass through snow to heat shingles whatever their color. In direct sunlight, snow is more likely to pass directly into the atmosphere by sublimation rather than melting so the angle of the roof relative to sun and shading are the important factors.

Air flow in an attic has less effect on heat transfer from the roof to the attic than commonly thought because most of that heat is in the form of infra red light energy that heats the floor without heating the air. The attic air eventually gets heated by the warm floor but at that point the heat transfer has occurred and the insulation in the floor is what must slow the heat transfer to the living space.

The essential purpose of attic ventilation is the removal of moisture although in humid climates introducing outside air can create moisture problems. The easiest solution is to include the attic space as part of the conditioned space of the house by insulating the rafters rather than the attic floor.

We like to find homes built in the late '70's or early 80's that have been owner occupied for 30+ years with a long term owner now moving on to condo or assisted living. We refurb them and rent them out. Sometimes we find one a year, sometimes we find a couple that fit our criteria and price.

In almost every case we need to somehow attend to the roof; venting, insulation, roofing, rain goods.

I can't speak to the science of it, our roofing guy does it, but there are calculations that decide how much soffit vent to ridge vent, how much insulation given your climate, etc.

In everyone of these homes that we have increased soffit venting, added soffit baffles and blown in insulation and installed continuous ridge venting what I can say is the "livability" of the home has demonstrably increased.

No more stuffy, no more stale. They just "feel" better.
Perhaps one of the experts can put the why of that into words, but to me its worth looking into if you are doing a re-roof.