If you really care about your painting and the impact it has on those who view it you will tackle the inside as well as the outside inmost things your do. It can be as simple as a single coat of colour. You could try a little glad wrap or sea sponge sponging, or use the product called soft flock for a velvet finish in a jewel box. On natural timber items that are to be used to store foods such as sugar canisters, potato bins..these I would leave in their natural state, even though most paint we use is labeled non toxic. Non toxic doesn't mean food safe. Keep that in mind if you choose to paint the inside of these. Read the label!
The finishing of your work is very important. All sides or areas of the piece deserve attention. They should all be sealed against the elements and finished against the elements. Never leave the back of a piece not done. It is not only unsightly but also unhealthy for your painting. If you don't finish the back, time and the elements will attack your painting from behind and ruin the work in years to come. Do the back to match the front: Same basecoat colour or contrast colour. Apply as much varnish here as on the front too!