This is an oft-recurring question in this forum. A search for "white lines" will likely find multiple threads on the subject.
The miniscule while lines of actually overlapping objects (not necessarily of supposedly abutting path edges) is an artifact of Illustrator's anti-aliasing display. You can easily check such artifacts by merely zooming in. Does the white sliver grow as you zoom? If not, it doesn't exist. Be aware that the same anti-aliasing artifacts are often more evident in Acrobat or Reader. So merely saving as a PDF is not a sure way to check whether the unwanted white actually exists.
Best thing to do, though, is simply understand the construction of your drawing. Select the associated paths. Are they actually overlapping objects, or are they the result of instant-gratification commands which try to automate "perfectly" abutting shapes, like auto-tracing, path combinations (so-called Pathfinders in Illustrator), or flood-fill tools (like Illustrator's Live Paint)?
But the question of whether unwanted white gaps (long called "slivers" in the printing trade) will "show up in print" is, well, another question.
When designing for print, you must always think in terms of inks, not just colors. Even if the path objects are actually overlapping in your drawing program, slivers can, and quite often do, occur in print when the colors involved do not have any inks in common. It occurs because presses are mechanical devices which lay down one ink at a time, and mechanical devices are not perfect. Mechanical linkages have play. Paper jogs as it flies through the rollers. Even humidity causes paper to shrink or expand. So separate inks never lay down in absolutely perfect alignment. This is what the matter of "trapping" is all about. To accommodate a reasonable amount of miss-alignment ("miss-registration"), colors involving dissimilar inks are either slightly outset ("spread") or inset ("choked") to create a small measure of overlap. Nowadays, this is typically done by yet another imperfect software automation at the printing house. Or it can be semi-automated in the design software by setting automatic trapping values in the file. Or, it can be deliberately drawn by the designer by insetting or offsetting the associated paths and manually assigning overprinting.
In such threads, some advisors will offhandedly tell you "don't worry about it; the printer will do it for you." But especially in the case of something you are calling a logo, that is quite often not the case. For example, the undefined "red" and "blue" colors in your screenshot may have an amount of one or more component inks in common to sufficiently reduce the noticeability of slivers due to slight miss-registration. Those might be either cyan or magenta or black percentages, assuming your design is to be offset printed in four-color process (CMYK). But properly prepared logos are commonly used for many more purposes than just CMYK.
Even in offset printing, a blue and red logo is likely to be printed in just two spot-color inks. If so, the two inks will not share any component at all.
The same logo file may be used to imprint garments. A screen-printed T-shirt will probably be printed in two spot-color inks. Screen printing typically requires more trapping than offset, and is quite likely not going to be automatically generated in a screen printing shop.
The same logo file may be sent to a sign shop to be cut from sign vinyl. Vinyl cutters are quite accurate, and vinyl is more dimensionally stable than paper, but the separate pieces of different vinyl colors usually have to be assembled by hand. So generous overlaps are common when preparing logo artwork because most businesses need signs.
So in other words, even though the white hairlines you seen in your artwork are probably just anti-aliasing screen display artifacts, that doesn't necessarily mean that slivers will not occur in real-world applications, especially for a logo.
JET
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