As the error says, PHP requires you to set a default timezone. This is actually quite a common error message when setting up a new server (or local development environment) with PHP.
You need to find your php.ini file on the server (ask your host where it is, or ask them to edit it for you). Find this part:
; Defines the default timezone used by the date ...
After working with EllisLab support on this issue, it turned out to be a rogue config item in the database for "server_offset" config item. It was set to -1140 for some reason.
They were kind enough to decode, unserialize, modify, re-serialize, and re-encode the variable for us and that fixed the problem.
Here's the response regarding the resolution from ...
Sound like the theme files didn't get uploaded to the production server fulling. I suggest you delete the Solspace Calendar theme directory which should be located at /themes/third_party/solspace_calendar (or something like that)... then reupload the theme directory. If that doesn't get it working, also reupload the module files.