Hello.
I suppose, miniBB should run well on PHP 5.5.x, but I couldn't be precise about 5.6. Most of the servers we're doing support for, are still on 5.4, many websites would require the total re-coding even for 5.5. That's why people prefer to stay with the older version. Also, you'd need to switch the mySQL driver, if you run it on 5.5 (older mySQL is deprecated starting from this version, you need to plug-in the mySQLi, which is now default for miniBB).
I see now the PHP dev.team is working out the 7th version already, so I'm not sure if it's truly worth to go through the older updates... not sure why your hoster took such a decision. As I know, except deprecating mySQL driver, these versions do not contain anything specific. So, miniBB should run well even with PHP 5.6, as it doesn't contain something specific, too... | 
