Not too long ago many websites used 14px font size as a fairly universal norm, but usability studies have shown that this size doesn’t make it easy for all users to interact with the content.
The most recent ‘advice’ I read about recommended font size said “at least 16px” and I tend to agree. Your text needs to be easily legible to as many users as possible, and since I hit my late 40s I have definitely struggled with the 14px size!
Bring it up to 16px and presto, it become less of a task to read. Make it a point or two bigger still and my brow starts relaxing – I no longer need to squint at the little squiggles on my screen and wonder what it’s supposed to say. I’d say I am not alone in this.
Readability is important. Drop below 14px and I would suggest Google starts paying attention to where you are using it. Drop below 12px and there could well be a UI / UX related negative effect on your rank. I would guess that the scope for size in the positive direction might be a lot more forgiving.
Aside from smallest font size, the logical progression of H element sizes is expected too, with H1 at the largest size, and progressively smaller down to a size similar to your smallest <p> size. H2 therefore should logically be smaller than H1 or you may be sending the wrong signals to users.