Just to add to amon's answer, here's are a couple of relevant definitions from Article 4 of the UK GDPR (emphasis added):
‘personal data’ means any information relating to an identified or identifiable natural person** (‘data subject’); an identifiable natural person is one who can be identified, directly or indirectly, in particular by reference to an identifier such as a name, an identification number, location data, an online identifier or to one or more factors specific to the physical, physiological, genetic, mental, economic, cultural or social identity of that natural person;
‘processing’ means any operation or set of operations which is performed on personal data or on sets of personal data, whether or not by automated means, such as collection, recording, organisation, structuring, storage, adaptation or alteration, retrieval, consultation, use, disclosure by transmission, dissemination or otherwise making available, alignment or combination, restriction, erasure or destruction;
So, what matters is whether the data you are collecting falls within the definition of personal data. If there is no personal data, then there is no processing for the purposes of the GDPR (the bolded part above).
There are two ways that data can be personal data: (1) it relates to an identified person; (2) it relates to an identifiable person.
You're not collecting names, so that rules out the first case. Your question is less clear about the second case. In the title you've written that you might collect "potentially personally identifiable information". What matters is whether it is possible to identify the person from the data that you collect.
You therefore have two options. Either operate outside the scope of the GDPR altogether by not collecting personal data, or comply with the GDPR.
If someone asks for the data to be removed, and you do not hold identifiable information, then there is no need for compliance. If you do hold identifiable information, and the data subject identifies themself (by reference to that information) in the request then you need to comply.