I just got an copyright infringement notice that says my app infringes on the usage policies of the services my app uses. 

However, this mail does not come directly from the company that runs the website I pull information from.
The mail comes from a company that has some paid apps in the same category on the app store (mine is free), so they obviously want me gone. 

My question is, can I just ignore this mail until I hear directly from the company that runs the website? Judging from their (the company that contacted me) website, they have no connection what so ever to the company that I fetch information from.

Be proactive and contact the company who runs the service you are using, and ask them if they are ok with your app being available on the app store, and with the way it uses their services. This approach has benefits: If you ignore the email from this competing company, they might inform the service owners of your app, which might lead them to ask you to take it down. If you approach them directly they might take a more lenient view as it shows you are willing to comply with their terms and not sneak something by them (even if that's what you've been doing so far).