You mean trademarks, not copyrights. Copyrights are for creative works like songs, books, films, sculpture, plays,. etc. Trademarks are logos and words that designate the source of products ot goods. The fabric sold it fabric stores with trademarks on it such as that of LA Dodgers is sold with a license for your PERSONAL use, not for COMMERCIAL sale.
The reason you'd be infringing the trademarks of the owners of those logos is that customers who see your diaper cakes (whatever that is) and baby wipe cases might think that they're official products made by or licensed by LA Dodgers products, with all the producct quality that that affilation implies, and not by you, whose quality may be just fine but is unknown to consumers. It's that CONSUMER CONFUSION that's the essence of trademark infringement.
If you use the LA Dodgers team colors and advertise your products using their tram name, same problem, those team colors are part of the Dodgers' "trade dress," and customers won't know that these products are from you.
The upshot is that you can't trade on other people's TM rights.
PLEASE READ THIS BEFORE YOU COMMENT, EMAIL ME OR PHONE ME. I'm only licensed in CA. This answer doesn't make me your lawyer, and neither do follow-up comments and/or emails and/or phone calls --- we need an actual agreement confirmed in writing before any attorney-client relationship is formed. Please don't expect me to respond to your follow-up queries. This answer doesn't constitute legal advice, and shouldn't be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.
Q: Is it illegal to buy fabric from a store w/ copyright logos such as the Dodgers, etc. and use it on crafts that I make and sell?
R: Yes. Read the responses and all the comments to the same question and its responses here: http://goo.gl/5OJXd
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
Yes it is illegal. This is a trademark law problem. You cannot use the logos associated with the Dodgers because you are attempting to profit from trademark rights held by others. Further, your use of the trademark would confuse consumers. You also cannot use the colors and call it a "Dodger-themed diaper cake". The Dodgers have a trademark in their name---further, even using the color scheme can violate "trade dress" rights held by the Dodgers. You should know that major league baseball and individual teams have become increasingly aggressive in enforcing their IP rights, such as trademarks and trade dress. The risk of a law suit is very high if you proceed with your plan.
Of course, if you want to pursue this, you cold always seek permission from the Dodgers. It is possible that you could obtain a license to use these rights for a reasonable license fee or royalty. If you want to pursue this matter, you should retain IP counsel to assist you in attempting to obtain a license. But you should know that the Dodgers have no obligation to give you a license--they could decide that they do not want you to use their IP rights.