It entirely depends on what those requirements were and whether you still fulfill them. Some of the criteria are more permanent than others (e.g., having a PhD from a top whatever uni, which is forever, vs. salary or professional experience within the last x years, which might change since the initial application).
The gold card is pretty much completely independent of whether you find a job here. Either you meet the requirements at the time of application or you don’t, and having a job in Taiwan vs. not having a job in Taiwan per se doesn’t really matter, unless it helps you meet the salary or other requirements.
If you don’t still meet the requirements for the gold card and want to keep working that job after the open work permit expires, I suppose you’d have to switch to a regular employment-based ARC and have the company apply for a new work permit etc.
You might also want to look into the rules for switching to an APRC (see here, and especially my question and the responses starting here). I don’t know your full life story of course, but if I remember correctly you had the gold card (a three-year one?) for ages before you actually moved to Taiwan? So I’m not sure whether this is a possibility for you at the moment.