We started using a covered litter box when we found our little one wasn't sitting down to pee, and was pee-ing all over the wall behind him!
He got used to it straight away, door on and everything. But yes, you should be able to take the door off to start with.
The only downside was that because the smells were contained within, cleaning it was pretty horrible. Also, for some reason, the litter didn't clump as well as it did in a non-covered box. We assume it has something to do with the restricted air flow. Instead it turned to a sort of mushy clay, which made it much harder to clean than before.
Overall tho, still suited our needs better than the other boxes as we'd much rather scrape clay off the box than wash pee off the wall!

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