In short, yes, but there are several variables that tie into this issue, so let's address a few!
Most woodworkers have encountered a scenario early on in their adoption of this great hobby where they have finally completed their first beautiful coffee table. They will have made seamless joints, and sanded every inch of the table up through countless grits until their project had a soft velvet feel. Finish and sealer were applied perfectly even with no runs or sags, and every single step in their coffee tables for dummies book was followed to a tee. They will bask in their feelings of accomplishment for a couple weeks until they walk into their living room and find that the piece of beautiful furniture they created now looks more like a half pipe suitable for Tony Hawk. But, why? How?
As I mentioned, there are several factors which affect the longevity of projects such as these. From the orientation of tabletop boards to the application of stains and sealers, there are numerous factors to keep an eye on throughout any project to ensure it stays straight and stable for years to come. Staining and sealing unseen areas of projects is just one of them.
Before we even get to finishing your projects, there is something to monitor during assembly which can greatly affect the outcome of any project, especially tabletops, that is wood grain. Wood grain is gorgeous and can really give a project some unique personality, but it can also be your worst enemy. Pay attention to the orientation of the end grain on any boards you plan to turn into a tabletop. On most boards you will notice a kind of semi-circle pattern formed from the tree's growth rings. As wood contracts or expands from moisture, it is then somewhat "guided" in its movements by these grain patterns. The wood will often cup or twist in the direction of the grain pattern. If all the boards in the tabletop have this pattern facing in the same direction, whether that be up or down, the likelihood that the tabletop will warp, or twist, is extremely high. From my experience, orienting these boards to have the grain patterns alternating every board can help mitigate warps and twists in the future.
Moisture is another important variable to monitor when attempting to avoid warps and twists in your projects. Often when buying wood from a supplier, especially a big box store, the moisture content in the wood does not match your local area. If this wood is cut and assembled into a project prior to the boards having a chance to expand or contract to the moisture in your air, then imperfections in your work are almost always going to rear their ugly heads. I always recommend allowing newly purchased wood to sit at your home, out of the weather, for seven to ten days prior to creating projects with them.
Lastly, to the question of whether or not the underside of a wood table must be stained, again, yes! I know that it can be easy to ignore those blind spots on projects that no one will ever see or notice, but it is pertinent to apply the same stain and sealers to inconspicuous spots of a project as the exposed sides. The reason for this is, yet again, moisture! If all sides of the wood are not finished evenly, then these areas of the project will not handle water or moisture evenly. This uneven contraction and expansion will certainly cause issues in your project over time. Another recommendation to keep from having difficulty with finishing inconspicuous areas of your project would be to stain your precut boards prior to assembly. I hope this is able to help a couple of you and stain those bottoms!