Most adhesives don't stick well to inked surfaces. Hot melt tapes, in particular, do not perform well on digital print. Neither does glue or glue dots (which is why there is no ink on the glue tabs of shipping and tuck-top boxes).
We recommend using acrylic tape on digitally printed boxes. We don't recommend any particular brand over another, but a product like Scotch 3650 Long Lasting Storage Tape that includes "acrylic adhesive" in the specifications should do the trick. "Storage Tape" in general is a step in the right direction, but it's important to look for the type of adhesive used -- "rubber adhesive" for example (most common), will not stick to digital inks.
Sometimes temperature can affect the adhesion of the tape, as well. If you are assembling your boxes in a cold warehouse during the winter, it can be more difficult to get tapes to stick.
We suggest that you consider the design of any closure materials and shipping labels when designing your boxes - leave an unprinted space, for example, where the label can reside, or select a decorative closing tape and leave unprinted space for it.
Some customers put a dab of glue on the underside of the front flap on mailer boxes, in which case leaving the inner flap unprinted is a great solution. However, if box reuse is important to you, be aware that this also makes those flaps really hard to pull apart, which can lead to damage.
Some form of tape is required to assemble shipper boxes. Mailer boxes, tuck-top cartons, and folding cartons close completely and snugly without any tape. However, if you are shipping your contents in one of these box styles, we do recommend some form of closure to keep things safe in transit.