I nearly fell off my chair reading this.
The short answer to this is yes, like many other fashion brands/designers in the UK & elsewhere that would like everyone to be walking around wearing their products and displaying their name wherever possible. Bench, G-Star, Henleys, Adidas, Dolce & Gabbana, Firetrap and Diesel to name but a handful.
The two exceptions mentioned here are both purchased in support of the band/team and should only be worn as such, they have nothing to do with fashion, unless the only reason you buy a shirt from a fashion retailer is to support the shirt designer's lifestyle because you are particularly fond of how they carry out their work.
I wouldn't lose the logo altogether, maybe just relocate & resize it to the sleeve, shoulder or back of the neck and I would certainly take away the 'tees.com' part as it will prove extremely difficult to transform a web address into a brand name. By all means keep the URL on the label and/or tags.
Best of luck with your shop, will send a 'like' and follow on twitter.