The console is fully disabled when playing on a Survival mode save (from the beta). There is no way to enable it I've seen, either from the .ini settings or by changing the keyboard bindings or even the launch options. It seems that the console is only disabled in Survival mode — in Normal difficulties the console is available and usable.
How can I re-enable the console in Survival mode?
To note, the Fallout 4 PC version is considerably very buggy. There are numerous spots where players can get stuck; console commands make getting out of these "traps" possible. Without console commands, it may be possible to get stuck and never get out.
Best Answer
Update:
A mod has been released that lets you enable survival mode multiple times. There is no .esp file, so no workaround is needed to load the mod. Now you can disable survival mode, use the console, and then re-enable survival. Note that all the survival stats (hunger, thirst, etc) will be reset.
Bethesda has disabled the console for the beta Survival mode as can be seen here and here (from an official Bethesda source).
There is currently no known workaround. Since it is disabled intentionally as a feature to prevent cheating, it is likely hard-coded and not changeable through modifying config files, as that would defeat the purpose.