Recently we have had a number of calls about arborvitae(cedars) that have split open from the last heavy snow. In answer to your questions, yes, you should remove the snow. As I write it is raining so that will do most of the work for you.
When you are done shaking off the snow (I like to use a broom) you should get in next to the plant and tie the main branches up. When tying up the branches you should use soft wide material. I often recommend nylons. The nylons are soft and will not cut into the skin of the trunk. Be creative; look around the house for wide soft material.
You may have to tie it up in one or two spots. This will greatly help the plant recover its original form. If this girding up is not done the plant may not recover its shape on its own.
I hope this is helpful.