I have been fortunate enough to be able to live in Belgium for 9 years and during this time I studied bobbin lace making and visited many exhibits of lace in Ghent and Bruges. I found out that care of lace is not difficult but it is a method that should be observed or one's priceless antiques or vintage pieces of dentelle (lace) will erode and degrade more quickly than if care is not observed.
I am NOT an expert on types of lace or their origin, however, while visiting the different exhibits and listening to my Belgian bobbin lace teacher, I have learned that one must not EVER use soap on their lace. Instead, one needs to rinse it in lukewarm water, do NOT wring out; instead place the lace on a piece of absorbent cotton cloth (towel will do) and press out the excess with a sponge then let air dry. After it is dry, use an iron with a press cloth and press the lace, you may have added a small amount of liquid starch during the rinsing process and if this is the case, press lightly on the iron but still continue to use a press cloth.
After this has been done, storing the cloth is fairly simple. Ensure that you have a place that is out of direct light/excessive humidity/excessive heat and have a box that is very clean. It is better to use an airtight container but not thoroughly necessary. Have tissue paper handy but please try to use acid free paper as it will help control the degradation of your lace and it should be more safe for years to come.
Handmade lace is an heirloom that should be passed down not only from person to person but from civilization to civilization so that all may have inspiration from it's beauty.
I hope this helps you with your treasures.