Is not awaiting for a promise an anti-pattern?

Question

I'm having an application where I have a function which exectues a task, but that task needs to do follow up, but not related to the task, so shouldn't be awaited. What are the potential dangers/drawbacks of not awaiting for a promise to be fulfilled/rejected?

For example we have an API handler, and we want to log that the user called the endpoint with this and that headers, etc. If I await for that log, the response will be slower with the execution time of the logging.