As far as I am aware both these answers are incorrect, but as I am not a lawyer let me quote the World Intellectual Property Organization (part of the UN):
Photos of trademarks
Unlike copyright law, trademark law as such does not restrict the use of a trademark in a photograph. What it does forbid is the use of a trademark in a way that can cause confusion regarding the affiliation of the trademark owner to the image. If consumers are likely to mistakenly believe that the trademark owner sponsored a photograph, then there may be trademark infringement. For example, if a Nike logo was visible on the t-shirt worn by the boy in our photo-shoot scenario, this could be seen as an attempt to appropriate consumer goodwill associated with the Nike trademark. So, caution is required if photographing someone wearing or consuming a trademarked product.
Source: IP and Business: Using Photographs of Copyrighted Works and Trademarks
So the basic idea is that if the Washington Redskins would not want to be associated with the Biden campaign, they could file a lawsuit claiming that the photo suggests there might be an affiliation between the campaign and the sport organization. In other words, the question one has to ask is: Is there a chance we might benefit in any way from the goodwill associated with the trademark?
The way it was explained in a copyright course1 I watched was that:
- you don't need to worry about a McDonald's in the background of your photo
- you need to be careful where you use a general photo of a specific McDonald's
- and taking a photo of a political candidate in front of a McDonald's is not acceptable without permission2
The biden campaign claimed that
A campaign aide told Fox News the logo was removed from the photo because it is "copyrighted" and claimed that such a step is "a very common practice on campaigns."
To my knowledge that are no actual copyright concerns here, but - just like with the aforementioned course - trademark matters often get covered in the same setting as copyright laws, so I have a very easy time believing that this is 'a very common practice on campaigns'. It's incredibly unlikely that for this specific photo the Washington Redskins would have actually claimed that Biden was benefiting from the goodwill associated with their brand (especially considering all the controversy surrounding them), but it's a completely believable general policy. The interesting thing is that they didn't care as much about postings on social media compared to more traditional channels, but this is in line with what I have seen in many companies and organizations.
1 - This course was trying to generalize international law in a way that content producers won't get in trouble anywhere rather than exclusively explain US law.
2 - The example didn't use a 'political candidate' explicitly, but something along the lines of a recognizable public figure who is not just getting a burger