The best way to fight against acidity in the soil is by adding limestone to it.
As long as the lawn is getting plenty of light under the
tree, and keep it reasonably clean of needles it should do fine.
Most people do not put lime down heavy enough to really make a difference however.
To lower the acidity one notch on the ph scale you will need to add about 30 lbs of limestone per 1000 square feet.
The best type of limestone to get is the pelletized type. It has already been broken down and reformed into pellets.
It is much cleaner then using the old fashioned barn lime. The best time to do this is in the fall.
Hope this helps!