First, it's important to realise that a round box and lid will become oval as their moisture content changes, so they may need to be turned upto 90 degrees to get them aligned properly before they will fit.
Rough turning and leaving for several months before finishing may help, both in drying and relieving stress in the wood. In practice I have not found this to be a complete solution to the problem.
Choice of wood may be critical. Look for a wood that shrinks very little, or has similar shrinkage characteristics in both tangential and radial directions. Some of the lowest North American T/R ratios (1.2 - 1.3) are yellow birch, southern magnolia, eastern hophornbeam. Some of the highest are beech, black cherry, American elm, sugar maple, up in the 1.9 - 2.2. I have to admit that most of the boxes I have tried in the past were maple, so maybe I will have to try again with yellow birch or hophornbeam.
Treating the wood to limit movement might be worth trying. Rough turn and leave to dry, then soak in a finish like danish oil before finish turning. There may also be mileage in trying one of the solutions that some bowl turners use for green wood, such as LDD or alcohol, or a proprietary product like pentacryl. I'm not sure though how well these will work on dry wood.
Another thing to consider is the climate you turn in. It might be worth turning boxes only when the RH is mid range, maybe 60%, rather than at an extreme like 30% or 90%.
|Turning Boxes, Fine Woodworking DVD|
Richard Raffan shows step-by-step the techniques and tricks to make elegant turned boxes with perfectly fitting lids. You’ll see firsthand how to do the work, the tools, the techniques, and the subtle rhythm of each process. 55 minutes.View sample video clip.
|Turning Boxes with Richard Raffan, revised|
Revised and updated full-length study of box-turning. When it comes to turning, there is no greater master to learn from than Richard Raffan. Here in Turning Boxes, Raffan reveals the tricks you need to know when turning boxes, from the cutting and seasoning of the wood to finishing the piece.