The same thing seems to have happened two years ago public-profile-url causing error, but the approved solution on that question is not solving this problem.
Any connections requests requesting the field 'public-profile-url' throws the following error:
https://api.linkedin.com/v1/people/~/connections:(public-profile-url)?start=0&oauth2_access_token=<token>
<error>
<status>500</status>
<timestamp>1491329264595</timestamp>
<request-id>8IJSBKJHQR</request-id>
<error-code>0</error-code>
<message>Internal API server error</message>
</error>
If this is just an issue with their API, I'd love to get LinkedIn's attention on the matter.
Edit: Interestingly, I found a specific Linkedin connection that is causing the failure (by adjusting the 'start' and 'count' args). It's probably not wise to post that here, but I can provide that information to LinkedIn staff, if need be.
Edit 2: I found replacing 'public-profile' with 'site-standard-profile-request' does function as a workaround. It's not exactly the same thing, but close enough if your user is already using oauth