Can I keep using my old academic affiliation for publishing?

When you are really not affiliated to the institution anymore, you should not put it on your publications. Certainly not, when the work you are publishing has not been initiated during your stay at the institution.

Many institutions have the possibility to offer you a 'zero hours' contract. In this scheme you remain affiliated to the institution, and can use their facilities (e.g., email address, etc), but do not receive any monetary compensation.

If you agree to such an arrangement with your previous institution, you obviously can still use the affiliation on your publications.

I have a colleague in urban studies who continues to publish research even though she is not currently in academia and runs a non-profit. She lists the non-profit she directs as her affiliation on publications. In her case, it appears impressive that she is a "practitioner-scholar." I don't know what your exact discipline is, but if your current job is okay with you publishing your work while employed there, why not list your employer? That doesn't affect the quality of your work. You can also credit your former indtitution in the suthor's note or author's bio, if there is one. Hope this helps!

It considerably depends whether the research you are publishing/presenting was done at your former institution or not.

Case 1: the research was done at your former institution

Your former institution supported your research (office, computer, lab, ...). The number of publications is an important metric for research institutions when they are evaluated. If you did your research at that institution, it is fair to list it as the affiliation of you or of one of the co-authors. Thus, the publication is counted as this institution's publication.

However, your former institution (head of department or similar) needs to be informed about the publication and should be asked in advance, whether it is desired to indicate the institution as your affiliation.

Case 2: the research was done after you left your former institution

In the second case, your former institution should not be mentioned because there is not connection between your current research and that institution.

However, if you are still using resources of that institution (e.g. a high performance computing cluster), you should discuss it with the responsible person there.