I've had success removing silkscreened logos from guitar gig bags and other cordura fabric type luggage.
Use any generic spraycan paint remover that shoots foam onto the work. Let foam sit for a few minutes to bite in. then wipe off vigorously with a rag. repeat until most of it is gone. Then quickly launder.
This might work for a cotton or polyester T-shirt too.
note: do at your own risk.

There are varous processes that pass as "silkscreening" featuring a variety of ink formulations... with or without heat bonding, and other variations. -- all resulting in an illustration or logo graphic applied to aparrel, luggage, or fabric.

There's no rule that says a guy can't try to change his T shirt design.
Worst that can happen is a vomit colored smear left on the front of the shirt.
Then again, the paint remover spray might just work to totally eradicate the previous design and leave the fabric base color intact, after laundering.