I've encountered dozens, maybe hundreds of problems where it was actually something that would matter. Here's one example. Consider you need to process lots of requests, that require making multiple requests that take a long time to finish (like 10-30 seconds) and you need to do it nearly in real-time. Given there's no sane multi-threading in PHP as well, you end up building your own parallelism support, which requires running lots of workers. Now, every of these workers does some pre-processing, makes a requests and waits for multiple damn seconds, to get the result back, make some computations and get to the next job. Consequently, to handle the same number of requests/sec you need much more workers, which has a lot of overhead, which results in needing more servers.
Sometimes you can use CURL to just make multiple similar requests in parallel and call it a day, but in practice I rarely could make use of it (consider that service you need to call might require some fucked up protocol, like SOAP, so you best option would be to rewrite SOAP client using CURL with parallel requests, which is not fun at all).
At that point you really start thinking some (any?!) other language would be a better fit for the application, but you have tons of very domain-specific business logic written in PHP that you'd need to rewrite for that app, as well as a team of PHP-developers that are not necessarily ready to learn whatever you have in mind, sooo...