For ChatGPT 3, the current date is inserted into a long pre-prompt, along with instructions like "this is a conversation between an AI chatbot and a human" plus "be nice" and "be truthful", which are part of the attempts to frame the next-word-predicting engine at the core of ChatGPT as a chatbot. OpenAI confirm this approach in their general GPT documentation.
Inherently, the core of ChatGPT - the GPT large language model - is not a chatbot. It has some resemblance conceptually to an image inpainting system — it predicts text that is likely, given preceding text.
I expect the same is true of ChatGPT 4, but have not seen any confirmation of this. It is feasible in principle to alter architecture of the bots to have them reference external data sources, but I believe that for the current date, a pre-prompt will still be in use.