People have different storage preferences for truffles, but most of them have the following things in common. Keeping them cool, relatively dry by absorbing excess moisture, and checking them daily.
The tips provided here are what I currently believe to be the best way to store truffles, however remember it is generally accepted that truffles are best used within 10 days of harvest. Although they "may" last up to 3 weeks, with poor care it can be only a few days.
A ripe truffle is generating a lot of volatile aromas that we want to keep inside the truffle until we use it. An airtight container is preferred, although you will likely discover they aren't as well sealed as you thought when the smell permeates the surroundings anyhow.
I prefer to use a plastic, airtight container, with a firmly clipped lid. Ensure that the container has no "plastic smell" even when brand new. Most of the "dishwasher/microwave safe" types seem odour free.
Some people use a glass jar with a clip lid and seal, and these are also good. The reason I prefer plastic is it is less heat conductive than glass, so less condensation forms on the walls, drying out the truffle less.
Place your truffle in the container with a fresh piece of unperfumed kitchen towel or facial tissue. This is to absorb a little of the moisture escaping the truffle, and should be checked/changed daily. If it's noticeably damp to the touch, wipe the container drier with it and replace with a dry one.
Place the container in a cool location, I use the vegetable crisper portion of the fridge, rather than the main shelves, the further insulation from the rest of the fridge definitely helps. -This observation has proven popular and is now also recommended by a number of truffle sellers.
Daily inspection is important. A ripe, refrigerated truffle is still respiring, and requires the small amount of fresh air provided by opening the container daily.
If your truffle gets too moist in the container a fine white "fur" will develop on the outer skin. This most often happens when the truffle is not inspected daily, releasing moisture and admitting fresh air.
If this happens to your truffle and it's slight, you can try gently washing it, (a fine brush helps) pat dry with paper, and allow to air dry for a while. Alternatively, you could peel off the affected surface and discard (or use in truffle vinaigrette). Too moist & too long will likely result in a slimey truffle, you could see if the inside is still usable.
A solution to white furred truffles used in France is to then brush the truffle with cognac (or vodka) This sterilises the truffle surface to a certain extent, but it will also alter the aroma's generated. Flavours should remain when cooked appropriately.
Infusing eggs, rice, cheese,...
You will discover the impressive power the truffle perfume has to penetate anything nearby. This opens a great deal of potential for cooking.
EGGS: The shell of a raw egg presents no obstacle to a ripe truffle, and it will easily infuse the egg with its aroma. Experiment with what suits with truffle to egg ratio, but dont try to stretch a truffle too far if you want to taste it readily. It will still need a tissue or little rice for moisture control. A simple infused egg, poached, on sourdough is a surprisingly effective dish.
RICE: I recommend you do not attempt to infuse rice for flavour, it withdraws too much moisture from the truffle and it's questionable that any aroma remains in the cooking rice. For risotto use parmesan instead as below.
A few spoons of rice are an alternative to paper tissue for absorbing moisture.
PARMESAN CHEESE: A surprising one, but the two intense flavours complement each other nicely, and truffled parmesan grated in and over a rissoto is superb. The granular structure of the parmesan allows the aroma to penetrate, and the cheese holds onto the truffle aromas quite well. Once again, despite its initial intensity, dont expect the "truffled" aspect to last more than a few days once the truffle is removed from the container.
BUTTER and OIL: Not really an infusion technique, Truffle butter is a great way to use any trimmings, small pieces etc. but this will only keep a few days before the taste deteriorates. Likewise any olive oil infusions you may prepare.
Most (if not ALL) commercial truffle oil has synthetic flavour added to it, (or even porcini mushroom powder). Due to their complexity, real truffle flavours are not stable in storage. None of the above will hold the aromas for very long once separated from the truffle.
[Disclaimer: many factors affect the storage and use of truffles, I provide the best information I can to help truffle lovers everywhere, but take all credit and no responsibility for anything that results from the use of this information]