When you see a print magazine on the shelf, obviously the cover page varies from month to month, however, enough stays constant that you can recognize the same magazine from one month to the next and that's how it should be - enough difference to show flair and variation, and avoid stagnation, but similar enough to provide consistency and recognition of brand.
It's no different online. It's still a balance. Vary the content and tweak the layout a little to make it different from one month to the next, but also to keep enough consistency to identify the brand. If the templates from one month to the next do not provide that base level branding and identity, then you'll likely lose readers.
So really in answer to your question, it depends just how different the templates are?
I'd suggest mocking something up that has the following (and get reader opinions):
a) the same content in two or three different templates and comparing the difference. If they look very different, chances are pretty high that the templates your considered are too different to take varying content from month to month and yet retain identity.
b) different content using the same template and comparing the similarity. If they look way too similar, the templates probably do need to be changed on an alternating basis (goto step c). If they really don't look very similar at all, you'll probably not want to make the situation worse by introducing varying templates.
c) different content in several different templates and comparing the difference. If they still seem to retain an identity between templates, chances are you'll be ok. If they look like completely different magazines, the templates are too different - reconsider.
And here's another consideration that makes the online experience a bit different:
Presumably, for many reasons, you are not going to remove the previous month's content as soon as the next month's content comes online. So you need to consider about how users arrive at a page (especially from google), and how they move from one month's cover page to another.
For example, can they link from Septembers issue to October's? Can they change the url from mymag.com/sep2011 to mymag.com/oct2011. You'll be wanting to make it very easy for someone coming in from google on an old issue to get at the current issue.
So, having viewed and digested the format of one month's cover, then the greater ease, (and the fewer in-between pages) with which the user can move from one month's cover page to the next, the less resistance you want to put in the readers way of being able to scan and decode the format of another month's page for the same magazine.
For the user's sake: make it all about the content, not about the packaging.