It is a bit labour intensive, but might be quicker than waiting for support to tell you which object custom field limit you have exceeded. That said, you will still need to ask support to increase the applicable limit if you can't delete any existing custom fields.
Create a new blank Development Org
Install the managed package there
For each object that gets custom fields installed, note how many the managed package requires.
Setup > App Setup > Customize > [Object. E.g. Opportunities] > Limits.
Compare this limit with the available capacity in your professional edition org.
Note 1: Not all managed packages have independent limits from the installing org.
Note 2: It might be quicker to scan the limits using
Just change pod to match your server instance and change the type query string parameter as required.
To clarify one point that I didn't see covered in the other answers:
Managed packages are only exempt from object, tab, and app limits IF the organization is part of the ISV program and the app is publicly posted on the app exchange (which requires passing security review). Documentation here. If the package is not listed on the app exchange it'll still affect your limits.
Also, sometimes something goes wrong with Salesforce and the owner of the managed package has to raise a support case to have the latest version of their package correctly not affect limits.