If an organization is currently heavily invested into customizations built for CiviCRM 4.6 LTS, is it possible for them to continue using the LTS version of CiviCRM when PHP 5.6 EOL? (31 Dec 2018) and an upgrade to PHP 7+ is necessary?
This kind of question is hard to answer for an open source project like CiviCRM :-) There is no 'owner' of the product that can tell you this. Nor is there a group of paid developers that will be able to continue supporting this. Considering that I think it is an impossible question to answer, and the wrong one to ask at this point in time. It would have to be one of the considerations when starting with heavy customization.
Having said that, I have worked in closed source situations where this kind of thing was promised by sales manager but never materialised :-). "No we will continue to support this version as long as you want to use it" and then a year later "we really urge you to upgrade as we will no longer support this version".
As an individual and member of the CiviCRM community I would not be interested in keeping a 4.6 version alive in combination with PHP 7. It would probably mean changing the 4.6 to a new PHP 7 compliant subversion, and then keeping it alive. That would leave me hardly any space for making a living. But that is solely as an individual with none of the organizations I work with facing this problem. It is only my personal opinion, it does not mean it is not possible or will never happen.
If there is a large group interested in doing this however you might be able to fund it. Or find partners that are willing to take this on. So I would really urge others to answer your question too!
Skvare was considering supporting 4.6 on php 7. See this link