I have an old compost pile that consists of leaves, lawn clippings and hay/chicken waste. Can I use lime to help break down the pile to a usable compost for my garden?
If it is an old pile it might finished compost. If the pieces are still too large try running a tiller thru it. The lime will not break it down but will only change the PH. Hope this helps.
If it is old yet barely-composted, your problem is that it has too much high-Carbon material compared to the amount of Nitrogen in there (which sounds likely, from what you say of its composition) and/or that it is too dry.
So maybe remix it with more high-N material (and stop adding hi-C material) and/or keep it thoroughly moist?
Lime will not make it compost any faster, in fact too much lime can retard composting.