Any such shrine would have to be sponsored entirely by voluntary donations, and make donors and visitors fully aware that it was neither sponsored by, nor endorsed by, Major League Baseball.
Provided those conditions were met, though, I'm not sure how Major League Baseball could effectively react; I don't think they'd have a meaningful trademark case (because of the explicit statements of non-endorsement) or could use other methods to shut the 'shrine' down. The real test for the shrine would be whether or not people took enough interest to make it worthwhile.