Does membership of academic honour societies carry any professional weight / recognition?

Does membership of honour society carry any professional weight?

No, not in general. At a very early stage in your career, for example when applying to graduate school, it may be useful as a quick indication that you have received high grades. However, the grades themselves are more meaningful than the honor society membership, and in any case grades only matter so much. It's worth mentioning the honor society on your CV, but it won't make a big difference.

Once you are more than a few years past undergraduate studies, neither grades nor honor societies matter.

In general, I think honor societies (including Golden Key) carry very little weight. One exception is Phi Beta Kappa, which (unlike Golden Key) does not require paying a fee to be inducted. I would be suspicious of any honor society that requires payment for membership.

Listing "prestigious" societies like Phi Beta Kappa or Tau Beta Pi is perfectly acceptable. While it probably won't do a whole lot of good, it also won't do any harm. Do avoid listing any societies where you have to pay dues or pay to be inducted: they are a double-whammy since they give the perception that 1) you are padding your CV with inconsequential honors, and 2) are exhibiting poor judgment by paying to do so.