First off, I'm not a lawyer yet, so none of this information should be taken as legal advice. It is educational only. For further guidance you should see an attorney.
DON'T let any of this scare you. Your initial post is very vague, so I have no idea why you'd be taking off the tags, but nothing is going to happen if you're just using cheap tees for your designs if it is for your own personal use or to make things for friends. Just be aware that if t-shirt design becomes a source of significant income, you'll want to talk to an attorney and set up an actual business relationship with a shirt producer. This all presumes that you're in the US. The UK is similar. I have no idea about other countries.
Technically it is violation of the Lanham Act called "reverse passing off". This is opposed to "passing off" which would be selling your goods as someone else's--- saying something you produced is another company's product. And yes, their tag is a trademark and your new one, even if unregistered is also a trademark. A trademark is basically anything that is indicative of source.
1-8 Trademark and Unfair Competition Deskbook § 8.02 [8] describes it as follows-
"A different form of unfair competition has been found to occur when a purchaser of goods removes the original trademark, substitutes another trademark of the purchaser's choosing, and then offers the product for resale. Such conduct has been termed ''reverse passing [or palming] off.'' Since the original trademark owner made its profit on the initial sale, the damage to it from such resales is not at once apparent. One court described the injury in this manner:
In reverse palming off cases, the originator of the misidentified product is involuntarily deprived of the advertising value of its name and of the goodwill that otherwise would stem from public knowledge of the true source of the satisfactory product. The ultimate purchaser is also deprived of knowing the true source of the product and may even be deceived into believing that it comes from another source."