Most people know that you can't copy the logo of a competitor in your field: aside from being unethical and a possible cause for legal action, it could also very well be a mark against the credibility of your company.
But this even applies for entities that aren't in the same business, or even in business at all, another reason your custom made signs should be original.
The Baltimore Sun recently reported on a dispute between Maryland senator Stephen Hershey and well-known chocolate company Hershey's. The latter has built up a very familiar brand presence with the use of a block font on a brown, chocolate-esque background.
As this source describes, the dispute is actually more than a decade old, stemming back to Senator Hershey's 2002 Commissioner campaign, when he first created the logo.
In the text of the lawsuit, the company asserts that this is an infringement of their copyright and "trade dress," but the senator claims that he was following the proper procedure and created his signs with permission.
Law360's news story on the subject featured comments from the company identifying the situations in which they feel that Hershey is free to use his name.
This does not include "adopting a design that uses the Hershey trade dress beyond any manner permissible under law and which defendants previously agreed not to use."
When a company starts investigating custom screen printing techniques, they should consider how to get their company more exposure without stepping on any toes. Just because you don't know that there's an infringement doesn't mean that there isn't one.