The easiest is to just cut it out with a sharp blade. Relatively safe if you're careful, though obviously you could damage the shirt if you're careless. Some large companies (e.g. Threadless) do it that way, though it can be pretty unprofessional looking if you try to do it quickly rather than accurately.
The easiest, safest, and most professional way is to outsource the work to a screenprinter, seamstress, etc. - but obviously that's not the cheapest way if cost is an issue.
The tag is sewn in during manufacture, so unpicking it isn't entirely trivial. I'm inclined to think anyone who needs to ask how to do it doesn't have the skills to do a neat job themselves (I know I would outsource it personally, but I haven't tried to do it myself so I may be overestimating the finesse required). Which pretty much leaves just cutting it out. If you cut too deep you'll cut the threads and leave a hole in the shirt, so you pretty much need to leave a stub. So your obvious choices are leave a slight stub and save some money, or pay someone to do it properly and eat into your profit margins.
Neither option is actually as bad as it sounds in my opinion.
I personally prefere to remove the original tag and press my own (cut or print and cut transfer).
This requires removing the original tag, this can be done either by carefully cutting, with an x-acto, along the holes of the stitches (of the original tag) or unstitching the tag and a small part of the rib. This method still requires sewing back the unstitched part of the rib, but easyer than stitching back your tag. Using transfer as a tag avoids stocking tags (orders for tags are typicaly by the thousands), manage tag stock by sizes, and after all transfer pressing is our business...
I'm now thinking about pressing my logo over the T brand on the poliester tag, more straigh to the point. When I make a few samples I will question the forum to see what's their opinion.
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