If a website's hosting is being transferred and is not online in between for approx. 14 days - would this affect the organic search positions when it's finally transferred? When they've since disappeared entirely from the search results?
On search results you rank, when users click on links to your website, they will encounter a website down/ similar error. This will come up as 404s in WMT. However, 404s does not hurt your ranking. I can think of three ways in which this is going to affect your rankings:
When Google try to access your website, it won't succeed. So this will come up as an error in Google WMT. As a general dictum, the less the errors, the better (though I cannot give you any proof for this).
When users click on links to your website (from other websites), they will encounter similar errors. Most webmasters prefer that their outgoing links be error free, so it is likely that your backlinks may get removed or replaced (even though unlikely in 2 weeks).
An inaccessible website or a website full of errors provide a negative sign to users. These users may be prospective customers. So it is less likely that they will link to you, in future too.
You can overcome most of these problems by giving your visitors a static page and a clear message with date as to when you will be back. However, problem 1 will remain.